| L(s) = 1 | − 3-s − 1.87·5-s − 0.0631·7-s + 9-s − 5.45·11-s + 5.11·13-s + 1.87·15-s − 1.55·17-s − 1.69·19-s + 0.0631·21-s − 2.04·23-s − 1.48·25-s − 27-s + 0.725·29-s + 9.80·31-s + 5.45·33-s + 0.118·35-s + 10.7·37-s − 5.11·39-s − 1.79·41-s + 7.31·43-s − 1.87·45-s − 2.10·47-s − 6.99·49-s + 1.55·51-s + 9.70·53-s + 10.2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.838·5-s − 0.0238·7-s + 0.333·9-s − 1.64·11-s + 1.41·13-s + 0.484·15-s − 0.376·17-s − 0.389·19-s + 0.0137·21-s − 0.426·23-s − 0.296·25-s − 0.192·27-s + 0.134·29-s + 1.76·31-s + 0.950·33-s + 0.0200·35-s + 1.77·37-s − 0.818·39-s − 0.280·41-s + 1.11·43-s − 0.279·45-s − 0.307·47-s − 0.999·49-s + 0.217·51-s + 1.33·53-s + 1.38·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 5 | \( 1 + 1.87T + 5T^{2} \) |
| 7 | \( 1 + 0.0631T + 7T^{2} \) |
| 11 | \( 1 + 5.45T + 11T^{2} \) |
| 13 | \( 1 - 5.11T + 13T^{2} \) |
| 17 | \( 1 + 1.55T + 17T^{2} \) |
| 19 | \( 1 + 1.69T + 19T^{2} \) |
| 23 | \( 1 + 2.04T + 23T^{2} \) |
| 29 | \( 1 - 0.725T + 29T^{2} \) |
| 31 | \( 1 - 9.80T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 1.79T + 41T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 + 2.10T + 47T^{2} \) |
| 53 | \( 1 - 9.70T + 53T^{2} \) |
| 59 | \( 1 + 7.68T + 59T^{2} \) |
| 61 | \( 1 - 1.95T + 61T^{2} \) |
| 67 | \( 1 - 2.58T + 67T^{2} \) |
| 71 | \( 1 + 5.70T + 71T^{2} \) |
| 73 | \( 1 - 0.697T + 73T^{2} \) |
| 79 | \( 1 + 6.35T + 79T^{2} \) |
| 83 | \( 1 + 0.371T + 83T^{2} \) |
| 89 | \( 1 + 6.15T + 89T^{2} \) |
| 97 | \( 1 + 2.20T + 97T^{2} \) |
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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
−7.80567554715654943873311912745, −6.81395128682071298634250596698, −6.09929379268628493964129419175, −5.58506546510052056402175546106, −4.55925953712763899372555669704, −4.17464306601881153349478025096, −3.16054884121423235652396494563, −2.34331888466842330334538004584, −1.02653837842831810769595014793, 0,
1.02653837842831810769595014793, 2.34331888466842330334538004584, 3.16054884121423235652396494563, 4.17464306601881153349478025096, 4.55925953712763899372555669704, 5.58506546510052056402175546106, 6.09929379268628493964129419175, 6.81395128682071298634250596698, 7.80567554715654943873311912745
