| L(s) = 1 | − 3·9-s + 12·13-s − 36·17-s − 56·29-s − 60·37-s + 4·41-s + 86·49-s − 204·53-s − 148·61-s − 264·73-s + 9·81-s + 28·89-s − 48·97-s − 40·101-s + 172·109-s − 252·113-s − 36·117-s + 230·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 108·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 0.923·13-s − 2.11·17-s − 1.93·29-s − 1.62·37-s + 4/41·41-s + 1.75·49-s − 3.84·53-s − 2.42·61-s − 3.61·73-s + 1/9·81-s + 0.314·89-s − 0.494·97-s − 0.396·101-s + 1.57·109-s − 2.23·113-s − 0.307·117-s + 1.90·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.705·153-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01085902613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01085902613\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 86 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 230 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 530 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 1010 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 430 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 190 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 1346 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 102 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3130 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 430 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 10034 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 132 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1682 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 94 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.716018936452984202390891814254, −9.045492322141150931408124317744, −8.989618646708631818833942500363, −8.782253626936753878170281264989, −8.190557119119297016984229695489, −7.55994306026124732453743996103, −7.43623616154388132538336921207, −6.83767705572325181423463983435, −6.36640512535190446898946115511, −5.98583209974019983062895041069, −5.74519278107940721781054856408, −4.81696726017546119962620155191, −4.80584202155084393737432738159, −4.02103559248388976424761877455, −3.70758625064693816471739048699, −3.04519113977483184084028260975, −2.56292298281281345009573820355, −1.68828380870196673499368885279, −1.53169448199448096316653308641, −0.02942861356119341813152622712,
0.02942861356119341813152622712, 1.53169448199448096316653308641, 1.68828380870196673499368885279, 2.56292298281281345009573820355, 3.04519113977483184084028260975, 3.70758625064693816471739048699, 4.02103559248388976424761877455, 4.80584202155084393737432738159, 4.81696726017546119962620155191, 5.74519278107940721781054856408, 5.98583209974019983062895041069, 6.36640512535190446898946115511, 6.83767705572325181423463983435, 7.43623616154388132538336921207, 7.55994306026124732453743996103, 8.190557119119297016984229695489, 8.782253626936753878170281264989, 8.989618646708631818833942500363, 9.045492322141150931408124317744, 9.716018936452984202390891814254
