| L(s) = 1 | + 2-s + 4-s − 4.22·7-s + 8-s + 5.13·11-s − 3.16·13-s − 4.22·14-s + 16-s − 6.48·17-s − 19-s + 5.13·22-s + 7.56·23-s − 3.16·26-s − 4.22·28-s − 0.832·29-s − 4.51·31-s + 32-s − 6.48·34-s − 0.137·37-s − 38-s + 11.6·41-s + 2.51·43-s + 5.13·44-s + 7.56·46-s − 5.96·47-s + 10.8·49-s − 3.16·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.59·7-s + 0.353·8-s + 1.54·11-s − 0.878·13-s − 1.12·14-s + 0.250·16-s − 1.57·17-s − 0.229·19-s + 1.09·22-s + 1.57·23-s − 0.621·26-s − 0.798·28-s − 0.154·29-s − 0.810·31-s + 0.176·32-s − 1.11·34-s − 0.0226·37-s − 0.162·38-s + 1.81·41-s + 0.382·43-s + 0.774·44-s + 1.11·46-s − 0.869·47-s + 1.55·49-s − 0.439·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.441913258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.441913258\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4.22T + 7T^{2} \) |
| 11 | \( 1 - 5.13T + 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 17 | \( 1 + 6.48T + 17T^{2} \) |
| 23 | \( 1 - 7.56T + 23T^{2} \) |
| 29 | \( 1 + 0.832T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 + 0.137T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 2.51T + 43T^{2} \) |
| 47 | \( 1 + 5.96T + 47T^{2} \) |
| 53 | \( 1 - 0.225T + 53T^{2} \) |
| 59 | \( 1 + 5.39T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 4.11T + 67T^{2} \) |
| 71 | \( 1 + 3.82T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 3.93T + 97T^{2} \) |
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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
−7.32099966240337576191712517378, −6.96605402924763937548298463606, −6.41540425603527620442676737521, −5.87756376913086678243832051690, −4.86142866212559333207919586867, −4.20519562391212201786812656703, −3.54173208619207850515771218072, −2.80944422190270651768728525085, −2.00711274011848064251231028654, −0.66202374538604611070695406826,
0.66202374538604611070695406826, 2.00711274011848064251231028654, 2.80944422190270651768728525085, 3.54173208619207850515771218072, 4.20519562391212201786812656703, 4.86142866212559333207919586867, 5.87756376913086678243832051690, 6.41540425603527620442676737521, 6.96605402924763937548298463606, 7.32099966240337576191712517378
