| L(s) = 1 | − 2.44·2-s + 3-s + 3.99·4-s + 5-s − 2.44·6-s + 7-s − 4.89·8-s + 9-s − 2.44·10-s − 2.44·11-s + 3.99·12-s + 4.44·13-s − 2.44·14-s + 15-s + 3.99·16-s + 5.44·17-s − 2.44·18-s − 4.44·19-s + 3.99·20-s + 21-s + 5.99·22-s − 4.89·24-s + 25-s − 10.8·26-s + 27-s + 3.99·28-s + 10.3·29-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 0.577·3-s + 1.99·4-s + 0.447·5-s − 0.999·6-s + 0.377·7-s − 1.73·8-s + 0.333·9-s − 0.774·10-s − 0.738·11-s + 1.15·12-s + 1.23·13-s − 0.654·14-s + 0.258·15-s + 0.999·16-s + 1.32·17-s − 0.577·18-s − 1.02·19-s + 0.894·20-s + 0.218·21-s + 1.27·22-s − 0.999·24-s + 0.200·25-s − 2.13·26-s + 0.192·27-s + 0.755·28-s + 1.92·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.437627517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.437627517\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 17 | \( 1 - 5.44T + 17T^{2} \) |
| 19 | \( 1 + 4.44T + 19T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 0.101T + 31T^{2} \) |
| 37 | \( 1 + 3.89T + 37T^{2} \) |
| 41 | \( 1 - 5.44T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 8.44T + 47T^{2} \) |
| 53 | \( 1 + 0.550T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 0.651T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 + 4.34T + 71T^{2} \) |
| 73 | \( 1 + 5.34T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 3.10T + 97T^{2} \) |
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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
−8.133288390915537116001746799587, −7.48474869758769243789137942707, −6.68524985391441048377856406940, −6.08821985361615007504236008191, −5.21571380858449291456234993993, −4.15128666075013073623805933854, −3.09045076073766406151469504376, −2.39924919137177860482013826785, −1.53490301174187662401944632193, −0.806809820769470084639737874661,
0.806809820769470084639737874661, 1.53490301174187662401944632193, 2.39924919137177860482013826785, 3.09045076073766406151469504376, 4.15128666075013073623805933854, 5.21571380858449291456234993993, 6.08821985361615007504236008191, 6.68524985391441048377856406940, 7.48474869758769243789137942707, 8.133288390915537116001746799587
