L(s) = 1 | + 8·2-s + 40·4-s + 56·7-s + 160·8-s + 5·9-s − 49·11-s + 52·13-s + 448·14-s + 560·16-s + 27·17-s + 40·18-s + 51·19-s − 392·22-s − 9·23-s + 416·26-s − 48·27-s + 2.24e3·28-s − 161·29-s + 38·31-s + 1.79e3·32-s + 216·34-s + 200·36-s + 16·37-s + 408·38-s + 547·41-s + 609·43-s − 1.96e3·44-s + ⋯ |
L(s) = 1 | + 2.82·2-s + 5·4-s + 3.02·7-s + 7.07·8-s + 5/27·9-s − 1.34·11-s + 1.10·13-s + 8.55·14-s + 35/4·16-s + 0.385·17-s + 0.523·18-s + 0.615·19-s − 3.79·22-s − 0.0815·23-s + 3.13·26-s − 0.342·27-s + 15.1·28-s − 1.03·29-s + 0.220·31-s + 9.89·32-s + 1.08·34-s + 0.925·36-s + 0.0710·37-s + 1.74·38-s + 2.08·41-s + 2.15·43-s − 6.71·44-s + ⋯ |
Λ(s)=(=((24⋅58⋅134)s/2ΓC(s)4L(s)Λ(4−s)
Λ(s)=(=((24⋅58⋅134)s/2ΓC(s+3/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
24⋅58⋅134
|
Sign: |
1
|
Analytic conductor: |
2.16330×106 |
Root analytic conductor: |
6.19283 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 24⋅58⋅134, ( :3/2,3/2,3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
128.1454744 |
L(21) |
≈ |
128.1454744 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | (1−pT)4 |
| 5 | | 1 |
| 13 | C1 | (1−pT)4 |
good | 3 | C2≀S4 | 1−5T2+16pT3−128T4+16p4T5−5p6T6+p12T8 |
| 7 | C2≀S4 | 1−8pT+2127T2−56354T3+1179224T4−56354p3T5+2127p6T6−8p10T7+p12T8 |
| 11 | C2≀S4 | 1+49T+3391T2+138994T3+5643870T4+138994p3T5+3391p6T6+49p9T7+p12T8 |
| 17 | C2≀S4 | 1−27T+2197T2−22226T3+35190538T4−22226p3T5+2197p6T6−27p9T7+p12T8 |
| 19 | C2≀S4 | 1−51T+56p2T2−1144575T3+184346126T4−1144575p3T5+56p8T6−51p9T7+p12T8 |
| 23 | C2≀S4 | 1+9T+19148T2−1050327T3+211262118T4−1050327p3T5+19148p6T6+9p9T7+p12T8 |
| 29 | C2≀S4 | 1+161T+62851T2+8580410T3+2244057936T4+8580410p3T5+62851p6T6+161p9T7+p12T8 |
| 31 | C2≀S4 | 1−38T+82751T2−2524314T3+3420131180T4−2524314p3T5+82751p6T6−38p9T7+p12T8 |
| 37 | C2≀S4 | 1−16T+92424T2+3890912T3+4714532798T4+3890912p3T5+92424p6T6−16p9T7+p12T8 |
| 41 | C2≀S4 | 1−547T+282444T2−78101601T3+24924328566T4−78101601p3T5+282444p6T6−547p9T7+p12T8 |
| 43 | C2≀S4 | 1−609T+210408T2−27097781T3+3443560014T4−27097781p3T5+210408p6T6−609p9T7+p12T8 |
| 47 | C2≀S4 | 1+254T+329955T2+78532974T3+46994253256T4+78532974p3T5+329955p6T6+254p9T7+p12T8 |
| 53 | C2≀S4 | 1−961T+826473T2−439333422T3+201643806858T4−439333422p3T5+826473p6T6−961p9T7+p12T8 |
| 59 | C2≀S4 | 1+230T+570243T2+184187238T3+147920246352T4+184187238p3T5+570243p6T6+230p9T7+p12T8 |
| 61 | C2≀S4 | 1−1393T+1287731T2−871191462T3+481233034676T4−871191462p3T5+1287731p6T6−1393p9T7+p12T8 |
| 67 | C2≀S4 | 1−579T+478955T2−66957006T3+73599298802T4−66957006p3T5+478955p6T6−579p9T7+p12T8 |
| 71 | C2≀S4 | 1−680T+1526320T2−718747160T3+834788988798T4−718747160p3T5+1526320p6T6−680p9T7+p12T8 |
| 73 | C2≀S4 | 1−1321T+1890170T2−1517221359T3+1160323729322T4−1517221359p3T5+1890170p6T6−1321p9T7+p12T8 |
| 79 | C2≀S4 | 1+55T+544628T2−9199713T3+329780786422T4−9199713p3T5+544628p6T6+55p9T7+p12T8 |
| 83 | C2≀S4 | 1−2733T+4979865T2−5812678818T3+5207004649132T4−5812678818p3T5+4979865p6T6−2733p9T7+p12T8 |
| 89 | C2≀S4 | 1+1233T+2665102T2+2131465291T3+2654755754554T4+2131465291p3T5+2665102p6T6+1233p9T7+p12T8 |
| 97 | C2≀S4 | 1−1250T+2495976T2−1844678942T3+2573508001070T4−1844678942p3T5+2495976p6T6−1250p9T7+p12T8 |
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L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−7.17974575182505976891922135176, −6.82030525931740493629307419425, −6.49500516102182579860576941327, −6.44514597265337479662479540840, −6.01663987694777599169674614651, −5.68531253569788687703482767345, −5.63790132492773667237855687565, −5.34631264018484191107742989911, −5.24181917168181349055555506252, −4.92368756401968252689892952692, −4.88549531679258987808338625536, −4.41379863714528606624197143969, −4.36595360976470844219781337698, −3.99830065172604267297895393791, −3.62918893561899683872168337741, −3.53351675285351490675953357109, −3.42830441525640553306252229339, −2.66335058184802715694427739823, −2.39552003818372709361648200977, −2.25912339245938537004282696603, −2.10490838248663253492119317506, −1.71305215942496138245145684718, −1.17554651769401414081471723671, −0.883822737783078875389563975974, −0.73932665550596750286870195681,
0.73932665550596750286870195681, 0.883822737783078875389563975974, 1.17554651769401414081471723671, 1.71305215942496138245145684718, 2.10490838248663253492119317506, 2.25912339245938537004282696603, 2.39552003818372709361648200977, 2.66335058184802715694427739823, 3.42830441525640553306252229339, 3.53351675285351490675953357109, 3.62918893561899683872168337741, 3.99830065172604267297895393791, 4.36595360976470844219781337698, 4.41379863714528606624197143969, 4.88549531679258987808338625536, 4.92368756401968252689892952692, 5.24181917168181349055555506252, 5.34631264018484191107742989911, 5.63790132492773667237855687565, 5.68531253569788687703482767345, 6.01663987694777599169674614651, 6.44514597265337479662479540840, 6.49500516102182579860576941327, 6.82030525931740493629307419425, 7.17974575182505976891922135176