| L(s) = 1 | − 2·3-s + 5-s − 7-s + 9-s + 2·11-s − 2·15-s − 17-s − 4·19-s + 2·21-s + 23-s + 25-s + 4·27-s + 5·29-s − 7·31-s − 4·33-s − 35-s + 37-s + 5·41-s + 4·43-s + 45-s − 4·47-s − 6·49-s + 2·51-s + 53-s + 2·55-s + 8·57-s − 9·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.516·15-s − 0.242·17-s − 0.917·19-s + 0.436·21-s + 0.208·23-s + 1/5·25-s + 0.769·27-s + 0.928·29-s − 1.25·31-s − 0.696·33-s − 0.169·35-s + 0.164·37-s + 0.780·41-s + 0.609·43-s + 0.149·45-s − 0.583·47-s − 6/7·49-s + 0.280·51-s + 0.137·53-s + 0.269·55-s + 1.05·57-s − 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.33863713552207448455550431234, −6.55228109223835450493523747584, −6.25756103536008634102415441902, −5.55997320989322812714829090886, −4.82509934625970776995433914069, −4.13168796451309598686367977990, −3.14107270231141501387125620373, −2.16032911529567247610592035624, −1.09985239101026226455477258174, 0,
1.09985239101026226455477258174, 2.16032911529567247610592035624, 3.14107270231141501387125620373, 4.13168796451309598686367977990, 4.82509934625970776995433914069, 5.55997320989322812714829090886, 6.25756103536008634102415441902, 6.55228109223835450493523747584, 7.33863713552207448455550431234
