| L(s) = 1 | + 2-s + 1.18·3-s + 4-s − 5-s + 1.18·6-s − 7-s + 8-s − 1.59·9-s − 10-s − 0.587·11-s + 1.18·12-s − 2.40·13-s − 14-s − 1.18·15-s + 16-s + 6.36·17-s − 1.59·18-s − 0.642·19-s − 20-s − 1.18·21-s − 0.587·22-s − 1.33·23-s + 1.18·24-s + 25-s − 2.40·26-s − 5.44·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.684·3-s + 0.5·4-s − 0.447·5-s + 0.484·6-s − 0.377·7-s + 0.353·8-s − 0.531·9-s − 0.316·10-s − 0.177·11-s + 0.342·12-s − 0.667·13-s − 0.267·14-s − 0.306·15-s + 0.250·16-s + 1.54·17-s − 0.375·18-s − 0.147·19-s − 0.223·20-s − 0.258·21-s − 0.125·22-s − 0.278·23-s + 0.242·24-s + 0.200·25-s − 0.471·26-s − 1.04·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 3 | \( 1 - 1.18T + 3T^{2} \) |
| 11 | \( 1 + 0.587T + 11T^{2} \) |
| 13 | \( 1 + 2.40T + 13T^{2} \) |
| 17 | \( 1 - 6.36T + 17T^{2} \) |
| 19 | \( 1 + 0.642T + 19T^{2} \) |
| 23 | \( 1 + 1.33T + 23T^{2} \) |
| 29 | \( 1 + 6.07T + 29T^{2} \) |
| 31 | \( 1 + 3.09T + 31T^{2} \) |
| 37 | \( 1 + 3.09T + 37T^{2} \) |
| 41 | \( 1 + 4.99T + 41T^{2} \) |
| 43 | \( 1 + 6.71T + 43T^{2} \) |
| 47 | \( 1 + 8.98T + 47T^{2} \) |
| 53 | \( 1 + 0.800T + 53T^{2} \) |
| 59 | \( 1 - 8.29T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 7.30T + 67T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 1.30T + 89T^{2} \) |
| 97 | \( 1 + 6.78T + 97T^{2} \) |
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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
−7.920192381414578347825601523332, −7.21952810881673130706886276809, −6.44949299779713128576024860100, −5.48034136095028133000809357227, −5.08557723012126577001696372594, −3.83517741834683903227444233289, −3.41872681266262532071795943832, −2.67621927585329197284133239335, −1.68699646045825452648934940972, 0,
1.68699646045825452648934940972, 2.67621927585329197284133239335, 3.41872681266262532071795943832, 3.83517741834683903227444233289, 5.08557723012126577001696372594, 5.48034136095028133000809357227, 6.44949299779713128576024860100, 7.21952810881673130706886276809, 7.920192381414578347825601523332
