| L(s) = 1 | + 1.62·3-s + 5-s + 1.05·7-s − 0.374·9-s + 1.92·11-s − 5.16·13-s + 1.62·15-s − 0.487·17-s + 3.35·19-s + 1.71·21-s − 8.10·23-s + 25-s − 5.46·27-s − 5.06·29-s − 9.15·31-s + 3.12·33-s + 1.05·35-s − 2.78·37-s − 8.36·39-s − 9.74·41-s + 9.98·43-s − 0.374·45-s − 12.5·47-s − 5.88·49-s − 0.789·51-s − 8.21·53-s + 1.92·55-s + ⋯ |
| L(s) = 1 | + 0.935·3-s + 0.447·5-s + 0.399·7-s − 0.124·9-s + 0.581·11-s − 1.43·13-s + 0.418·15-s − 0.118·17-s + 0.769·19-s + 0.374·21-s − 1.68·23-s + 0.200·25-s − 1.05·27-s − 0.940·29-s − 1.64·31-s + 0.544·33-s + 0.178·35-s − 0.458·37-s − 1.34·39-s − 1.52·41-s + 1.52·43-s − 0.0558·45-s − 1.83·47-s − 0.840·49-s − 0.110·51-s − 1.12·53-s + 0.260·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 - 1.62T + 3T^{2} \) |
| 7 | \( 1 - 1.05T + 7T^{2} \) |
| 11 | \( 1 - 1.92T + 11T^{2} \) |
| 13 | \( 1 + 5.16T + 13T^{2} \) |
| 17 | \( 1 + 0.487T + 17T^{2} \) |
| 19 | \( 1 - 3.35T + 19T^{2} \) |
| 23 | \( 1 + 8.10T + 23T^{2} \) |
| 29 | \( 1 + 5.06T + 29T^{2} \) |
| 31 | \( 1 + 9.15T + 31T^{2} \) |
| 37 | \( 1 + 2.78T + 37T^{2} \) |
| 41 | \( 1 + 9.74T + 41T^{2} \) |
| 43 | \( 1 - 9.98T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 + 8.21T + 53T^{2} \) |
| 59 | \( 1 - 6.65T + 59T^{2} \) |
| 61 | \( 1 - 7.66T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 5.43T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 5.38T + 83T^{2} \) |
| 89 | \( 1 + 5.15T + 89T^{2} \) |
| 97 | \( 1 + 4.23T + 97T^{2} \) |
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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
−7.70367200842431650857473697731, −7.27260384900480107275230090277, −6.31724875950791081948211142343, −5.49183321623260799017248941819, −4.89152612288433592901537684336, −3.84096151373590789246927491420, −3.23996094957161974492260628412, −2.15747130424140685227181067600, −1.77591427713038857411955999872, 0,
1.77591427713038857411955999872, 2.15747130424140685227181067600, 3.23996094957161974492260628412, 3.84096151373590789246927491420, 4.89152612288433592901537684336, 5.49183321623260799017248941819, 6.31724875950791081948211142343, 7.27260384900480107275230090277, 7.70367200842431650857473697731
