| L(s) = 1 | − 2-s + 4-s + 1.61·5-s − 8-s − 1.61·10-s + 11-s + 16-s + 7.34·17-s + 1.17·19-s + 1.61·20-s − 22-s − 3.55·23-s − 2.39·25-s − 10.3·29-s + 6.34·31-s − 32-s − 7.34·34-s + 2.82·37-s − 1.17·38-s − 1.61·40-s − 4.94·41-s − 11.5·43-s + 44-s + 3.55·46-s − 6.78·47-s + 2.39·50-s − 8·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.721·5-s − 0.353·8-s − 0.510·10-s + 0.301·11-s + 0.250·16-s + 1.78·17-s + 0.268·19-s + 0.360·20-s − 0.213·22-s − 0.741·23-s − 0.479·25-s − 1.93·29-s + 1.13·31-s − 0.176·32-s − 1.25·34-s + 0.465·37-s − 0.189·38-s − 0.255·40-s − 0.772·41-s − 1.75·43-s + 0.150·44-s + 0.524·46-s − 0.989·47-s + 0.339·50-s − 1.09·53-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 | \( 1 - 1.61T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 7.34T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 3.55T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 6.34T + 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 + 4.94T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 6.78T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + 4.39T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 2.94T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 + 5.11T + 73T^{2} \) |
| 79 | \( 1 - 5.61T + 79T^{2} \) |
| 83 | \( 1 + 5.05T + 83T^{2} \) |
| 89 | \( 1 - 0.773T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−7.58706112006155892498588354682, −6.64802767496475947553437890589, −5.99440362039303322292340427587, −5.54224113410139393507225843392, −4.65303737481324065232243151987, −3.55821888774025826215802508992, −3.01080496136979650057679337074, −1.82692573456607302970413694887, −1.39126790606822179345833944557, 0,
1.39126790606822179345833944557, 1.82692573456607302970413694887, 3.01080496136979650057679337074, 3.55821888774025826215802508992, 4.65303737481324065232243151987, 5.54224113410139393507225843392, 5.99440362039303322292340427587, 6.64802767496475947553437890589, 7.58706112006155892498588354682
