| L(s) = 1 | + 11.7·3-s + 59.9·5-s + 169.·7-s − 104.·9-s − 546.·11-s + 132.·13-s + 706.·15-s − 1.73e3·17-s + 401.·19-s + 1.99e3·21-s + 882.·23-s + 464.·25-s − 4.09e3·27-s + 1.38e3·29-s − 5.53e3·31-s − 6.43e3·33-s + 1.01e4·35-s + 2.86e3·37-s + 1.55e3·39-s − 1.68e4·41-s + 2.33e3·43-s − 6.23e3·45-s − 1.70e3·47-s + 1.18e4·49-s − 2.04e4·51-s − 1.01e4·53-s − 3.27e4·55-s + ⋯ |
| L(s) = 1 | + 0.755·3-s + 1.07·5-s + 1.30·7-s − 0.428·9-s − 1.36·11-s + 0.217·13-s + 0.810·15-s − 1.45·17-s + 0.255·19-s + 0.986·21-s + 0.347·23-s + 0.148·25-s − 1.07·27-s + 0.306·29-s − 1.03·31-s − 1.02·33-s + 1.39·35-s + 0.344·37-s + 0.164·39-s − 1.56·41-s + 0.192·43-s − 0.459·45-s − 0.112·47-s + 0.703·49-s − 1.10·51-s − 0.497·53-s − 1.45·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 - 11.7T + 243T^{2} \) |
| 5 | \( 1 - 59.9T + 3.12e3T^{2} \) |
| 7 | \( 1 - 169.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 546.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 132.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.73e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 401.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 882.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.38e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.53e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.86e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.68e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.33e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.70e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.01e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.32e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.80e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.93e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.89e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.22e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.59e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.750855261782478428283794774185, −8.112924031982193842781940182991, −7.33372030693064800745719728117, −6.10088702655916301409411751543, −5.29156644565536558894616270074, −4.57864353570403884582085966451, −3.13079225145260560256071367493, −2.23558451898498484917229353284, −1.65472838300112486392394590978, 0,
1.65472838300112486392394590978, 2.23558451898498484917229353284, 3.13079225145260560256071367493, 4.57864353570403884582085966451, 5.29156644565536558894616270074, 6.10088702655916301409411751543, 7.33372030693064800745719728117, 8.112924031982193842781940182991, 8.750855261782478428283794774185
