| L(s) = 1 | − 2-s + 0.0343·3-s + 4-s − 1.14·5-s − 0.0343·6-s − 1.82·7-s − 8-s − 2.99·9-s + 1.14·10-s − 6.15·11-s + 0.0343·12-s + 3.42·13-s + 1.82·14-s − 0.0392·15-s + 16-s − 6.14·17-s + 2.99·18-s − 19-s − 1.14·20-s − 0.0626·21-s + 6.15·22-s + 2.52·23-s − 0.0343·24-s − 3.69·25-s − 3.42·26-s − 0.206·27-s − 1.82·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.0198·3-s + 0.5·4-s − 0.510·5-s − 0.0140·6-s − 0.689·7-s − 0.353·8-s − 0.999·9-s + 0.361·10-s − 1.85·11-s + 0.00991·12-s + 0.951·13-s + 0.487·14-s − 0.0101·15-s + 0.250·16-s − 1.49·17-s + 0.706·18-s − 0.229·19-s − 0.255·20-s − 0.0136·21-s + 1.31·22-s + 0.527·23-s − 0.00701·24-s − 0.739·25-s − 0.672·26-s − 0.0396·27-s − 0.344·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03445754257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03445754257\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 0.0343T + 3T^{2} \) |
| 5 | \( 1 + 1.14T + 5T^{2} \) |
| 7 | \( 1 + 1.82T + 7T^{2} \) |
| 11 | \( 1 + 6.15T + 11T^{2} \) |
| 13 | \( 1 - 3.42T + 13T^{2} \) |
| 17 | \( 1 + 6.14T + 17T^{2} \) |
| 23 | \( 1 - 2.52T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 37 | \( 1 + 2.56T + 37T^{2} \) |
| 41 | \( 1 - 6.81T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 7.15T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 + 3.84T + 67T^{2} \) |
| 71 | \( 1 - 5.54T + 71T^{2} \) |
| 73 | \( 1 + 2.52T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 6.41T + 83T^{2} \) |
| 89 | \( 1 - 0.508T + 89T^{2} \) |
| 97 | \( 1 - 3.82T + 97T^{2} \) |
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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
−7.953623199762211574354186296107, −7.34526930242056838294505695253, −6.42354232416775750423332097465, −5.96174279692673891204786692976, −5.14542868319076347237724439643, −4.23679202110533437908626321273, −3.17769365412203538120787930996, −2.77293089993046464536680739004, −1.76560675486358333156277581810, −0.095300564398878658815920415479,
0.095300564398878658815920415479, 1.76560675486358333156277581810, 2.77293089993046464536680739004, 3.17769365412203538120787930996, 4.23679202110533437908626321273, 5.14542868319076347237724439643, 5.96174279692673891204786692976, 6.42354232416775750423332097465, 7.34526930242056838294505695253, 7.953623199762211574354186296107
