L(s) = 1 | + 0.513·2-s − 1.73·4-s + 3.39·5-s − 2.33·7-s − 1.91·8-s + 1.74·10-s − 6.00·11-s − 0.529·13-s − 1.19·14-s + 2.48·16-s − 3.47·17-s + 1.60·19-s − 5.88·20-s − 3.08·22-s + 4.78·23-s + 6.50·25-s − 0.271·26-s + 4.05·28-s − 0.564·29-s − 3.16·31-s + 5.11·32-s − 1.78·34-s − 7.92·35-s − 6.70·37-s + 0.821·38-s − 6.50·40-s − 1.61·41-s + ⋯ |
L(s) = 1 | + 0.362·2-s − 0.868·4-s + 1.51·5-s − 0.883·7-s − 0.678·8-s + 0.550·10-s − 1.80·11-s − 0.146·13-s − 0.320·14-s + 0.622·16-s − 0.841·17-s + 0.367·19-s − 1.31·20-s − 0.656·22-s + 0.997·23-s + 1.30·25-s − 0.0533·26-s + 0.767·28-s − 0.104·29-s − 0.567·31-s + 0.903·32-s − 0.305·34-s − 1.33·35-s − 1.10·37-s + 0.133·38-s − 1.02·40-s − 0.252·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.535773953\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.535773953\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 223 | \( 1 + T \) |
good | 2 | \( 1 - 0.513T + 2T^{2} \) |
| 5 | \( 1 - 3.39T + 5T^{2} \) |
| 7 | \( 1 + 2.33T + 7T^{2} \) |
| 11 | \( 1 + 6.00T + 11T^{2} \) |
| 13 | \( 1 + 0.529T + 13T^{2} \) |
| 17 | \( 1 + 3.47T + 17T^{2} \) |
| 19 | \( 1 - 1.60T + 19T^{2} \) |
| 23 | \( 1 - 4.78T + 23T^{2} \) |
| 29 | \( 1 + 0.564T + 29T^{2} \) |
| 31 | \( 1 + 3.16T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 + 1.61T + 41T^{2} \) |
| 43 | \( 1 - 7.24T + 43T^{2} \) |
| 47 | \( 1 - 3.63T + 47T^{2} \) |
| 53 | \( 1 - 0.415T + 53T^{2} \) |
| 59 | \( 1 - 8.45T + 59T^{2} \) |
| 61 | \( 1 - 4.30T + 61T^{2} \) |
| 67 | \( 1 - 7.14T + 67T^{2} \) |
| 71 | \( 1 - 9.02T + 71T^{2} \) |
| 73 | \( 1 - 4.46T + 73T^{2} \) |
| 79 | \( 1 - 2.33T + 79T^{2} \) |
| 83 | \( 1 - 1.10T + 83T^{2} \) |
| 89 | \( 1 + 7.97T + 89T^{2} \) |
| 97 | \( 1 + 9.18T + 97T^{2} \) |
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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
−8.225386276550287900177053150352, −7.21368781180298497809808122400, −6.53914530557658893807897705493, −5.61553552850259880146385236642, −5.38340342139583167966612225215, −4.68738191964007298533059399920, −3.54480833273140935093317408990, −2.77836034833280582619996371436, −2.10555904282164254022306683295, −0.59130827969484103403420518458,
0.59130827969484103403420518458, 2.10555904282164254022306683295, 2.77836034833280582619996371436, 3.54480833273140935093317408990, 4.68738191964007298533059399920, 5.38340342139583167966612225215, 5.61553552850259880146385236642, 6.53914530557658893807897705493, 7.21368781180298497809808122400, 8.225386276550287900177053150352