L(s) = 1 | − 2-s − 2.93·3-s + 4-s + 2.93·6-s + 1.31·7-s − 8-s + 5.63·9-s − 0.258·11-s − 2.93·12-s + 5.87·13-s − 1.31·14-s + 16-s − 4.25·17-s − 5.63·18-s + 2.93·19-s − 3.87·21-s + 0.258·22-s + 8.51·23-s + 2.93·24-s − 5.87·26-s − 7.75·27-s + 1.31·28-s − 3.61·29-s + 3.95·31-s − 32-s + 0.760·33-s + 4.25·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.69·3-s + 0.5·4-s + 1.19·6-s + 0.498·7-s − 0.353·8-s + 1.87·9-s − 0.0780·11-s − 0.848·12-s + 1.63·13-s − 0.352·14-s + 0.250·16-s − 1.03·17-s − 1.32·18-s + 0.674·19-s − 0.846·21-s + 0.0551·22-s + 1.77·23-s + 0.599·24-s − 1.15·26-s − 1.49·27-s + 0.249·28-s − 0.672·29-s + 0.710·31-s − 0.176·32-s + 0.132·33-s + 0.730·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7545775620\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7545775620\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 + 2.93T + 3T^{2} \) |
| 7 | \( 1 - 1.31T + 7T^{2} \) |
| 11 | \( 1 + 0.258T + 11T^{2} \) |
| 13 | \( 1 - 5.87T + 13T^{2} \) |
| 17 | \( 1 + 4.25T + 17T^{2} \) |
| 19 | \( 1 - 2.93T + 19T^{2} \) |
| 23 | \( 1 - 8.51T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 41 | \( 1 + 8.89T + 41T^{2} \) |
| 43 | \( 1 - 5.61T + 43T^{2} \) |
| 47 | \( 1 + 5.57T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 9.57T + 59T^{2} \) |
| 61 | \( 1 - 0.380T + 61T^{2} \) |
| 67 | \( 1 + 5.57T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 2.51T + 73T^{2} \) |
| 79 | \( 1 + 7.69T + 79T^{2} \) |
| 83 | \( 1 - 6.17T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−9.252467086120254580781724185356, −8.541661701548113654853042062607, −7.57792937120389235420518032945, −6.67314643949743651285876319896, −6.24219752039969092829996505298, −5.28323230659025158824120521520, −4.62650136906483906964779689949, −3.35593204904791113041879714947, −1.66420630891156449353936369759, −0.76068170620473418817346201720,
0.76068170620473418817346201720, 1.66420630891156449353936369759, 3.35593204904791113041879714947, 4.62650136906483906964779689949, 5.28323230659025158824120521520, 6.24219752039969092829996505298, 6.67314643949743651285876319896, 7.57792937120389235420518032945, 8.541661701548113654853042062607, 9.252467086120254580781724185356