| L(s) = 1 | − 36·11-s + 88·19-s + 540·29-s + 408·31-s − 160·41-s − 470·49-s − 348·59-s + 372·61-s + 264·71-s + 1.09e3·79-s − 2.10e3·89-s − 2.42e3·101-s − 436·109-s − 1.69e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.25e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 0.986·11-s + 1.06·19-s + 3.45·29-s + 2.36·31-s − 0.609·41-s − 1.37·49-s − 0.767·59-s + 0.780·61-s + 0.441·71-s + 1.56·79-s − 2.50·89-s − 2.39·101-s − 0.383·109-s − 1.26·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.93·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.557039215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.557039215\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 470 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 18 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4250 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 1410 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 44 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21198 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 270 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 204 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 86906 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 80 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 128282 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 79650 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3990 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 174 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 186 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 491302 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 132 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 415630 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 548 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 901510 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1052 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1593022 T^{2} + p^{6} T^{4} \) |
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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
−9.136162193010788684800734842270, −8.403067907619076028860805646621, −8.395969035919889411742556903936, −8.000121568220148223626781858429, −7.74977217428101164807082945165, −6.96801319888474612578489728749, −6.71041816159655205056699468160, −6.47655899909137745016848401940, −5.91352818237706722953689235099, −5.39064350490598919691795901459, −5.01764003930457173514927059144, −4.57071191601871278805886525346, −4.41147812847254433616767844724, −3.52064180233799899614844974727, −3.10957093581264403132511884938, −2.61263777159922354249339581060, −2.46157359309648286394846649531, −1.32301069127202222771317500044, −1.10148430213982332647464912168, −0.37960386425764411261458211341,
0.37960386425764411261458211341, 1.10148430213982332647464912168, 1.32301069127202222771317500044, 2.46157359309648286394846649531, 2.61263777159922354249339581060, 3.10957093581264403132511884938, 3.52064180233799899614844974727, 4.41147812847254433616767844724, 4.57071191601871278805886525346, 5.01764003930457173514927059144, 5.39064350490598919691795901459, 5.91352818237706722953689235099, 6.47655899909137745016848401940, 6.71041816159655205056699468160, 6.96801319888474612578489728749, 7.74977217428101164807082945165, 8.000121568220148223626781858429, 8.395969035919889411742556903936, 8.403067907619076028860805646621, 9.136162193010788684800734842270
