L(s) = 1 | − 2-s + 4-s − 1.37·5-s − 7-s − 8-s + 1.37·10-s − 4·11-s + 1.37·13-s + 14-s + 16-s + 4.74·17-s + 4·19-s − 1.37·20-s + 4·22-s + 23-s − 3.11·25-s − 1.37·26-s − 28-s − 9.37·29-s + 6.74·31-s − 32-s − 4.74·34-s + 1.37·35-s + 2.62·37-s − 4·38-s + 1.37·40-s + 8.11·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.613·5-s − 0.377·7-s − 0.353·8-s + 0.433·10-s − 1.20·11-s + 0.380·13-s + 0.267·14-s + 0.250·16-s + 1.15·17-s + 0.917·19-s − 0.306·20-s + 0.852·22-s + 0.208·23-s − 0.623·25-s − 0.269·26-s − 0.188·28-s − 1.74·29-s + 1.21·31-s − 0.176·32-s − 0.813·34-s + 0.231·35-s + 0.431·37-s − 0.648·38-s + 0.216·40-s + 1.26·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 1.37T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 1.37T + 13T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 + 9.37T + 29T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 - 2.62T + 37T^{2} \) |
| 41 | \( 1 - 8.11T + 41T^{2} \) |
| 43 | \( 1 - 6.11T + 43T^{2} \) |
| 47 | \( 1 + 4.62T + 47T^{2} \) |
| 53 | \( 1 + 4.74T + 53T^{2} \) |
| 59 | \( 1 - 2.74T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 7.48T + 89T^{2} \) |
| 97 | \( 1 + 9.37T + 97T^{2} \) |
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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
−8.273103891265869413935274601959, −7.56201928784635752452427021030, −7.37333219103131871544848108594, −6.00469954452246217645829278501, −5.56828305232552346323215667902, −4.37793510505903826212730695620, −3.35411761774041151867754606391, −2.66909224231013193317197729225, −1.26750770067846259783786485162, 0,
1.26750770067846259783786485162, 2.66909224231013193317197729225, 3.35411761774041151867754606391, 4.37793510505903826212730695620, 5.56828305232552346323215667902, 6.00469954452246217645829278501, 7.37333219103131871544848108594, 7.56201928784635752452427021030, 8.273103891265869413935274601959