| L(s) = 1 | − 2-s + 4-s − 1.11·5-s − 0.851·7-s − 8-s + 1.11·10-s − 3.02·11-s − 5.45·13-s + 0.851·14-s + 16-s − 1.69·17-s + 4.75·19-s − 1.11·20-s + 3.02·22-s + 9.18·23-s − 3.76·25-s + 5.45·26-s − 0.851·28-s + 5.73·29-s − 7.53·31-s − 32-s + 1.69·34-s + 0.947·35-s − 11.1·37-s − 4.75·38-s + 1.11·40-s + 0.585·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.497·5-s − 0.321·7-s − 0.353·8-s + 0.351·10-s − 0.910·11-s − 1.51·13-s + 0.227·14-s + 0.250·16-s − 0.410·17-s + 1.09·19-s − 0.248·20-s + 0.644·22-s + 1.91·23-s − 0.752·25-s + 1.06·26-s − 0.160·28-s + 1.06·29-s − 1.35·31-s − 0.176·32-s + 0.290·34-s + 0.160·35-s − 1.82·37-s − 0.772·38-s + 0.175·40-s + 0.0914·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5785298446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5785298446\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 1.11T + 5T^{2} \) |
| 7 | \( 1 + 0.851T + 7T^{2} \) |
| 11 | \( 1 + 3.02T + 11T^{2} \) |
| 13 | \( 1 + 5.45T + 13T^{2} \) |
| 17 | \( 1 + 1.69T + 17T^{2} \) |
| 19 | \( 1 - 4.75T + 19T^{2} \) |
| 23 | \( 1 - 9.18T + 23T^{2} \) |
| 29 | \( 1 - 5.73T + 29T^{2} \) |
| 31 | \( 1 + 7.53T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 - 0.585T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 + 9.81T + 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 61 | \( 1 + 6.51T + 61T^{2} \) |
| 67 | \( 1 + 3.41T + 67T^{2} \) |
| 71 | \( 1 - 0.424T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 1.81T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 - 9.01T + 97T^{2} \) |
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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
−7.76901008605450265255246275912, −7.13862013147459489921841628510, −6.92677055294074111265295213513, −5.64033558879506992072619326607, −5.15505714943298522018615043506, −4.34689658609850994297768318164, −3.14875137615736429211962888611, −2.78021380669019100360163059577, −1.68249801865800730791929646538, −0.40696690239686631777860588534,
0.40696690239686631777860588534, 1.68249801865800730791929646538, 2.78021380669019100360163059577, 3.14875137615736429211962888611, 4.34689658609850994297768318164, 5.15505714943298522018615043506, 5.64033558879506992072619326607, 6.92677055294074111265295213513, 7.13862013147459489921841628510, 7.76901008605450265255246275912
