| L(s) = 1 | + 0.805·2-s − 1.35·4-s + 1.59·5-s + 1.80·7-s − 2.69·8-s + 1.28·10-s + 4.74·11-s + 2.50·13-s + 1.45·14-s + 0.531·16-s − 6.69·17-s − 19-s − 2.16·20-s + 3.81·22-s + 6.91·23-s − 2.44·25-s + 2.01·26-s − 2.44·28-s − 1.55·29-s + 4.96·31-s + 5.82·32-s − 5.38·34-s + 2.88·35-s + 1.44·37-s − 0.805·38-s − 4.31·40-s + 9.91·41-s + ⋯ |
| L(s) = 1 | + 0.569·2-s − 0.675·4-s + 0.714·5-s + 0.683·7-s − 0.954·8-s + 0.407·10-s + 1.43·11-s + 0.694·13-s + 0.388·14-s + 0.132·16-s − 1.62·17-s − 0.229·19-s − 0.483·20-s + 0.814·22-s + 1.44·23-s − 0.488·25-s + 0.395·26-s − 0.461·28-s − 0.288·29-s + 0.892·31-s + 1.02·32-s − 0.923·34-s + 0.488·35-s + 0.237·37-s − 0.130·38-s − 0.682·40-s + 1.54·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.125371227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.125371227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 0.805T + 2T^{2} \) |
| 5 | \( 1 - 1.59T + 5T^{2} \) |
| 7 | \( 1 - 1.80T + 7T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 - 2.50T + 13T^{2} \) |
| 17 | \( 1 + 6.69T + 17T^{2} \) |
| 23 | \( 1 - 6.91T + 23T^{2} \) |
| 29 | \( 1 + 1.55T + 29T^{2} \) |
| 31 | \( 1 - 4.96T + 31T^{2} \) |
| 37 | \( 1 - 1.44T + 37T^{2} \) |
| 41 | \( 1 - 9.91T + 41T^{2} \) |
| 43 | \( 1 - 6.27T + 43T^{2} \) |
| 53 | \( 1 - 5.98T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 + 6.34T + 61T^{2} \) |
| 67 | \( 1 + 3.95T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 1.50T + 79T^{2} \) |
| 83 | \( 1 - 4.08T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 7.85T + 97T^{2} \) |
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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
−7.888889257471980532762007591487, −6.94154105992473693655435746523, −6.13522556248588429821931506843, −5.93512531732510670708970158558, −4.75577383948035133379809419245, −4.46638843789118629619837836636, −3.72436599200842917850799398277, −2.74846172378852450580261683091, −1.76324899513353783262915883923, −0.851935577044853447376529928376,
0.851935577044853447376529928376, 1.76324899513353783262915883923, 2.74846172378852450580261683091, 3.72436599200842917850799398277, 4.46638843789118629619837836636, 4.75577383948035133379809419245, 5.93512531732510670708970158558, 6.13522556248588429821931506843, 6.94154105992473693655435746523, 7.888889257471980532762007591487
