| L(s) = 1 | + 2-s + 4-s + 2.64·5-s + 8-s + 2.64·10-s − 11-s − 4·13-s + 16-s + 3·17-s − 5.29·19-s + 2.64·20-s − 22-s − 2.64·23-s + 2.00·25-s − 4·26-s − 2·29-s − 4·31-s + 32-s + 3·34-s − 9.29·37-s − 5.29·38-s + 2.64·40-s − 9·41-s − 1.29·43-s − 44-s − 2.64·46-s − 11.9·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.18·5-s + 0.353·8-s + 0.836·10-s − 0.301·11-s − 1.10·13-s + 0.250·16-s + 0.727·17-s − 1.21·19-s + 0.591·20-s − 0.213·22-s − 0.551·23-s + 0.400·25-s − 0.784·26-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s − 1.52·37-s − 0.858·38-s + 0.418·40-s − 1.40·41-s − 0.196·43-s − 0.150·44-s − 0.390·46-s − 1.74·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2.64T + 5T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + 2.64T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 9.29T + 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 1.29T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 + 3.93T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 79 | \( 1 - 5.35T + 79T^{2} \) |
| 83 | \( 1 - 5.58T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 9.58T + 97T^{2} \) |
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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
−7.10526583277237411484371916517, −6.57701248164210012835587204339, −5.85317841156047568771201320891, −5.25832770121441114498776103726, −4.79588018584106298209245175258, −3.78526277519252135488512400289, −3.03379175806038509593832848952, −2.07583230112106834888351709169, −1.73059376129159953175002230519, 0,
1.73059376129159953175002230519, 2.07583230112106834888351709169, 3.03379175806038509593832848952, 3.78526277519252135488512400289, 4.79588018584106298209245175258, 5.25832770121441114498776103726, 5.85317841156047568771201320891, 6.57701248164210012835587204339, 7.10526583277237411484371916517
