| L(s) = 1 | − 0.618·2-s − 1.61·4-s + 5-s + 0.236·7-s + 2.23·8-s − 0.618·10-s − 4.23·11-s + 13-s − 0.145·14-s + 1.85·16-s + 1.47·17-s − 6.47·19-s − 1.61·20-s + 2.61·22-s + 4.47·23-s + 25-s − 0.618·26-s − 0.381·28-s + 29-s − 8·31-s − 5.61·32-s − 0.909·34-s + 0.236·35-s + 4.00·38-s + 2.23·40-s + 6·41-s − 6·43-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.809·4-s + 0.447·5-s + 0.0892·7-s + 0.790·8-s − 0.195·10-s − 1.27·11-s + 0.277·13-s − 0.0389·14-s + 0.463·16-s + 0.357·17-s − 1.48·19-s − 0.361·20-s + 0.558·22-s + 0.932·23-s + 0.200·25-s − 0.121·26-s − 0.0721·28-s + 0.185·29-s − 1.43·31-s − 0.993·32-s − 0.156·34-s + 0.0399·35-s + 0.648·38-s + 0.353·40-s + 0.937·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 8.23T + 47T^{2} \) |
| 53 | \( 1 - 6.47T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 0.472T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 6.47T + 83T^{2} \) |
| 89 | \( 1 + 7.94T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−9.166437029600229514373069297033, −8.516474787541335897742630655885, −7.84213691539561513094503197416, −6.90359754271810280952952978964, −5.73546016520287620052703915199, −5.05966910170309996443735000591, −4.14671601274682740769912554271, −2.90502396691828259351079837347, −1.60458100225051803101519010156, 0,
1.60458100225051803101519010156, 2.90502396691828259351079837347, 4.14671601274682740769912554271, 5.05966910170309996443735000591, 5.73546016520287620052703915199, 6.90359754271810280952952978964, 7.84213691539561513094503197416, 8.516474787541335897742630655885, 9.166437029600229514373069297033
