L(s) = 1 | + 2-s + 0.747·3-s + 4-s − 1.63·5-s + 0.747·6-s − 7-s + 8-s − 2.44·9-s − 1.63·10-s − 4.29·11-s + 0.747·12-s + 6.70·13-s − 14-s − 1.22·15-s + 16-s − 1.42·17-s − 2.44·18-s + 3.01·19-s − 1.63·20-s − 0.747·21-s − 4.29·22-s + 0.971·23-s + 0.747·24-s − 2.32·25-s + 6.70·26-s − 4.06·27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.431·3-s + 0.5·4-s − 0.731·5-s + 0.305·6-s − 0.377·7-s + 0.353·8-s − 0.813·9-s − 0.517·10-s − 1.29·11-s + 0.215·12-s + 1.85·13-s − 0.267·14-s − 0.315·15-s + 0.250·16-s − 0.346·17-s − 0.575·18-s + 0.691·19-s − 0.365·20-s − 0.163·21-s − 0.916·22-s + 0.202·23-s + 0.152·24-s − 0.464·25-s + 1.31·26-s − 0.782·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.655319608\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.655319608\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
good | 3 | \( 1 - 0.747T + 3T^{2} \) |
| 5 | \( 1 + 1.63T + 5T^{2} \) |
| 11 | \( 1 + 4.29T + 11T^{2} \) |
| 13 | \( 1 - 6.70T + 13T^{2} \) |
| 17 | \( 1 + 1.42T + 17T^{2} \) |
| 19 | \( 1 - 3.01T + 19T^{2} \) |
| 23 | \( 1 - 0.971T + 23T^{2} \) |
| 29 | \( 1 - 9.17T + 29T^{2} \) |
| 31 | \( 1 + 8.60T + 31T^{2} \) |
| 37 | \( 1 + 0.352T + 37T^{2} \) |
| 41 | \( 1 + 0.0784T + 41T^{2} \) |
| 43 | \( 1 - 7.08T + 43T^{2} \) |
| 47 | \( 1 - 8.20T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 1.43T + 59T^{2} \) |
| 61 | \( 1 + 3.40T + 61T^{2} \) |
| 67 | \( 1 + 5.22T + 67T^{2} \) |
| 71 | \( 1 - 9.88T + 71T^{2} \) |
| 73 | \( 1 - 2.30T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−7.962504214316696132475791489481, −7.49885310759436426624488286239, −6.56053952348408555860877992378, −5.79364292104055204103922371327, −5.33005795974165416974265485638, −4.24090767394585746260824642379, −3.58194661187217343891028874871, −3.01310993526618343562890082979, −2.19592352527649120087971158843, −0.73201285980955530997919618697,
0.73201285980955530997919618697, 2.19592352527649120087971158843, 3.01310993526618343562890082979, 3.58194661187217343891028874871, 4.24090767394585746260824642379, 5.33005795974165416974265485638, 5.79364292104055204103922371327, 6.56053952348408555860877992378, 7.49885310759436426624488286239, 7.962504214316696132475791489481