| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 3·9-s − 10-s − 11-s − 14-s + 16-s + 4·17-s + 3·18-s + 20-s + 22-s + 25-s + 28-s + 8·31-s − 32-s − 4·34-s + 35-s − 3·36-s − 4·37-s − 40-s − 10·41-s − 44-s − 3·45-s + 4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 9-s − 0.316·10-s − 0.301·11-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.707·18-s + 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.188·28-s + 1.43·31-s − 0.176·32-s − 0.685·34-s + 0.169·35-s − 1/2·36-s − 0.657·37-s − 0.158·40-s − 1.56·41-s − 0.150·44-s − 0.447·45-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.90670620278901, −13.40659999847501, −12.47583237924829, −12.33577202581439, −11.56430154365317, −11.46129070084126, −10.66363155351532, −10.31458842371395, −9.925455880703556, −9.322265665201021, −8.773027448750169, −8.433941585595002, −7.861732568924749, −7.562832686135370, −6.699850848499025, −6.386859431223407, −5.732876794215026, −5.266155715252293, −4.845809081208287, −3.974710558771848, −3.194931134916363, −2.866470575404318, −2.152710010353607, −1.524516630872817, −0.8349925625627975, 0,
0.8349925625627975, 1.524516630872817, 2.152710010353607, 2.866470575404318, 3.194931134916363, 3.974710558771848, 4.845809081208287, 5.266155715252293, 5.732876794215026, 6.386859431223407, 6.699850848499025, 7.562832686135370, 7.861732568924749, 8.433941585595002, 8.773027448750169, 9.322265665201021, 9.925455880703556, 10.31458842371395, 10.66363155351532, 11.46129070084126, 11.56430154365317, 12.33577202581439, 12.47583237924829, 13.40659999847501, 13.90670620278901
