| L(s) = 1 | − 3-s − 2.80·5-s + 4.80·7-s + 9-s − 1.46·11-s + 2.80·15-s − 2.44·17-s − 2.54·19-s − 4.80·21-s + 3.51·23-s + 2.85·25-s − 27-s + 1.85·29-s + 7.63·31-s + 1.46·33-s − 13.4·35-s − 4.55·37-s + 1.24·41-s − 2.38·43-s − 2.80·45-s − 12.8·47-s + 16.0·49-s + 2.44·51-s − 8.85·53-s + 4.10·55-s + 2.54·57-s − 2.17·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.25·5-s + 1.81·7-s + 0.333·9-s − 0.442·11-s + 0.723·15-s − 0.593·17-s − 0.583·19-s − 1.04·21-s + 0.733·23-s + 0.570·25-s − 0.192·27-s + 0.343·29-s + 1.37·31-s + 0.255·33-s − 2.27·35-s − 0.748·37-s + 0.194·41-s − 0.363·43-s − 0.417·45-s − 1.86·47-s + 2.29·49-s + 0.342·51-s − 1.21·53-s + 0.554·55-s + 0.336·57-s − 0.283·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2.80T + 5T^{2} \) |
| 7 | \( 1 - 4.80T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 + 2.54T + 19T^{2} \) |
| 23 | \( 1 - 3.51T + 23T^{2} \) |
| 29 | \( 1 - 1.85T + 29T^{2} \) |
| 31 | \( 1 - 7.63T + 31T^{2} \) |
| 37 | \( 1 + 4.55T + 37T^{2} \) |
| 41 | \( 1 - 1.24T + 41T^{2} \) |
| 43 | \( 1 + 2.38T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 + 8.85T + 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 + 7.82T + 61T^{2} \) |
| 67 | \( 1 - 3.58T + 67T^{2} \) |
| 71 | \( 1 - 8.83T + 71T^{2} \) |
| 73 | \( 1 + 7.69T + 73T^{2} \) |
| 79 | \( 1 - 4.02T + 79T^{2} \) |
| 83 | \( 1 - 0.652T + 83T^{2} \) |
| 89 | \( 1 - 6.29T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.59252904490280002911015152896, −6.88366889187667061264006806164, −6.14100759199415691748954335292, −4.96711698134447443500017922298, −4.83218458492577701461055795075, −4.16655739806712117005005788203, −3.18866982013216305274387279240, −2.08301381156695593002848589701, −1.15972287209929871956668103191, 0,
1.15972287209929871956668103191, 2.08301381156695593002848589701, 3.18866982013216305274387279240, 4.16655739806712117005005788203, 4.83218458492577701461055795075, 4.96711698134447443500017922298, 6.14100759199415691748954335292, 6.88366889187667061264006806164, 7.59252904490280002911015152896
