| L(s) = 1 | + 1.41·3-s + 2.80·5-s + 3.87·7-s − 0.992·9-s + 1.16·11-s + 3.17·13-s + 3.97·15-s + 6.44·17-s + 5.29·19-s + 5.48·21-s − 3.18·23-s + 2.86·25-s − 5.65·27-s − 6.33·29-s + 0.489·31-s + 1.64·33-s + 10.8·35-s + 6.85·37-s + 4.50·39-s − 9.56·41-s + 9.97·43-s − 2.78·45-s + 7.89·47-s + 8.00·49-s + 9.12·51-s − 12.5·53-s + 3.25·55-s + ⋯ |
| L(s) = 1 | + 0.818·3-s + 1.25·5-s + 1.46·7-s − 0.330·9-s + 0.349·11-s + 0.881·13-s + 1.02·15-s + 1.56·17-s + 1.21·19-s + 1.19·21-s − 0.663·23-s + 0.573·25-s − 1.08·27-s − 1.17·29-s + 0.0878·31-s + 0.286·33-s + 1.83·35-s + 1.12·37-s + 0.721·39-s − 1.49·41-s + 1.52·43-s − 0.415·45-s + 1.15·47-s + 1.14·49-s + 1.27·51-s − 1.72·53-s + 0.438·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.951818232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.951818232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 7 | \( 1 - 3.87T + 7T^{2} \) |
| 11 | \( 1 - 1.16T + 11T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 17 | \( 1 - 6.44T + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 + 3.18T + 23T^{2} \) |
| 29 | \( 1 + 6.33T + 29T^{2} \) |
| 31 | \( 1 - 0.489T + 31T^{2} \) |
| 37 | \( 1 - 6.85T + 37T^{2} \) |
| 41 | \( 1 + 9.56T + 41T^{2} \) |
| 43 | \( 1 - 9.97T + 43T^{2} \) |
| 47 | \( 1 - 7.89T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 3.57T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 - 0.781T + 67T^{2} \) |
| 71 | \( 1 + 7.01T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 3.20T + 79T^{2} \) |
| 83 | \( 1 + 2.05T + 83T^{2} \) |
| 89 | \( 1 + 7.35T + 89T^{2} \) |
| 97 | \( 1 - 3.04T + 97T^{2} \) |
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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
−7.73472712085086047548087450967, −7.56875337831453697837609680202, −6.22598337615185458208677816690, −5.65227649591597475547730555867, −5.29067392473961128458862009227, −4.19354862996114016593523421642, −3.40401221531135915974805392579, −2.60176193929038324392005681028, −1.68065109996465637967932672601, −1.24430849677380932670671040827,
1.24430849677380932670671040827, 1.68065109996465637967932672601, 2.60176193929038324392005681028, 3.40401221531135915974805392579, 4.19354862996114016593523421642, 5.29067392473961128458862009227, 5.65227649591597475547730555867, 6.22598337615185458208677816690, 7.56875337831453697837609680202, 7.73472712085086047548087450967
