L(s) = 1 | − 2-s + 2·3-s + 4-s − 1.38·5-s − 2·6-s − 7-s − 8-s + 9-s + 1.38·10-s + 5.09·11-s + 2·12-s − 4.47·13-s + 14-s − 2.76·15-s + 16-s + 1.61·17-s − 18-s − 2.47·19-s − 1.38·20-s − 2·21-s − 5.09·22-s + 4.76·23-s − 2·24-s − 3.09·25-s + 4.47·26-s − 4·27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 0.5·4-s − 0.618·5-s − 0.816·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.437·10-s + 1.53·11-s + 0.577·12-s − 1.24·13-s + 0.267·14-s − 0.713·15-s + 0.250·16-s + 0.392·17-s − 0.235·18-s − 0.567·19-s − 0.309·20-s − 0.436·21-s − 1.08·22-s + 0.993·23-s − 0.408·24-s − 0.618·25-s + 0.877·26-s − 0.769·27-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
good | 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 11 | \( 1 - 5.09T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 - 4.76T + 23T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 - 0.763T + 31T^{2} \) |
| 37 | \( 1 + 1.14T + 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 6.76T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 1.14T + 61T^{2} \) |
| 67 | \( 1 + 4.47T + 67T^{2} \) |
| 71 | \( 1 - 7.38T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 6.47T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 8.18T + 97T^{2} \) |
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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
−7.71555784961638560485211222898, −7.34625693616956343970197454813, −6.58451324468671066099994469358, −5.77354912803037655143694846055, −4.56268493270006655898099315577, −3.81676572380033380211835086335, −3.13055294437571548313448896068, −2.35814007416460580471004262159, −1.37959181240276743289808263272, 0,
1.37959181240276743289808263272, 2.35814007416460580471004262159, 3.13055294437571548313448896068, 3.81676572380033380211835086335, 4.56268493270006655898099315577, 5.77354912803037655143694846055, 6.58451324468671066099994469358, 7.34625693616956343970197454813, 7.71555784961638560485211222898