| L(s) = 1 | − 0.618·2-s + 3-s − 1.61·4-s − 0.381·5-s − 0.618·6-s + 7-s + 2.23·8-s + 9-s + 0.236·10-s − 1.61·12-s − 13-s − 0.618·14-s − 0.381·15-s + 1.85·16-s − 4.23·17-s − 0.618·18-s + 0.618·20-s + 21-s − 3.23·23-s + 2.23·24-s − 4.85·25-s + 0.618·26-s + 27-s − 1.61·28-s + 6.70·29-s + 0.236·30-s − 10.2·31-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 0.577·3-s − 0.809·4-s − 0.170·5-s − 0.252·6-s + 0.377·7-s + 0.790·8-s + 0.333·9-s + 0.0746·10-s − 0.467·12-s − 0.277·13-s − 0.165·14-s − 0.0986·15-s + 0.463·16-s − 1.02·17-s − 0.145·18-s + 0.138·20-s + 0.218·21-s − 0.674·23-s + 0.456·24-s − 0.970·25-s + 0.121·26-s + 0.192·27-s − 0.305·28-s + 1.24·29-s + 0.0430·30-s − 1.83·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 5 | \( 1 + 0.381T + 5T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 7.32T + 47T^{2} \) |
| 53 | \( 1 - 7.61T + 53T^{2} \) |
| 59 | \( 1 + 4.14T + 59T^{2} \) |
| 61 | \( 1 + 5.76T + 61T^{2} \) |
| 67 | \( 1 + 9.23T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 6.38T + 89T^{2} \) |
| 97 | \( 1 + 17T + 97T^{2} \) |
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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
−8.542123553517384569906677613959, −7.906802116845379933939033747114, −7.34609170748319579452977878051, −6.26391632122639790626963707869, −5.23340949401010683283726812076, −4.37789098463777976596906138270, −3.83352386693344944516989287109, −2.54918067573751394248045488241, −1.49748775095175458438348381906, 0,
1.49748775095175458438348381906, 2.54918067573751394248045488241, 3.83352386693344944516989287109, 4.37789098463777976596906138270, 5.23340949401010683283726812076, 6.26391632122639790626963707869, 7.34609170748319579452977878051, 7.906802116845379933939033747114, 8.542123553517384569906677613959
