L(s) = 1 | + 2-s + 4-s − 2.61·5-s − 5.13·7-s + 8-s − 2.61·10-s + 2.47·11-s + 3.80·13-s − 5.13·14-s + 16-s + 2.41·17-s + 5.05·19-s − 2.61·20-s + 2.47·22-s + 5.71·23-s + 1.81·25-s + 3.80·26-s − 5.13·28-s − 3.62·29-s − 4.02·31-s + 32-s + 2.41·34-s + 13.4·35-s − 1.49·37-s + 5.05·38-s − 2.61·40-s + 10.3·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.16·5-s − 1.94·7-s + 0.353·8-s − 0.825·10-s + 0.747·11-s + 1.05·13-s − 1.37·14-s + 0.250·16-s + 0.584·17-s + 1.16·19-s − 0.583·20-s + 0.528·22-s + 1.19·23-s + 0.363·25-s + 0.745·26-s − 0.970·28-s − 0.672·29-s − 0.723·31-s + 0.176·32-s + 0.413·34-s + 2.26·35-s − 0.245·37-s + 0.820·38-s − 0.412·40-s + 1.61·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.849112062\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.849112062\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + 2.61T + 5T^{2} \) |
| 7 | \( 1 + 5.13T + 7T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 - 3.80T + 13T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 23 | \( 1 - 5.71T + 23T^{2} \) |
| 29 | \( 1 + 3.62T + 29T^{2} \) |
| 31 | \( 1 + 4.02T + 31T^{2} \) |
| 37 | \( 1 + 1.49T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 0.257T + 43T^{2} \) |
| 47 | \( 1 - 7.24T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 - 3.42T + 59T^{2} \) |
| 61 | \( 1 + 9.61T + 61T^{2} \) |
| 67 | \( 1 + 3.26T + 67T^{2} \) |
| 71 | \( 1 + 7.55T + 71T^{2} \) |
| 73 | \( 1 - 3.93T + 73T^{2} \) |
| 79 | \( 1 - 0.964T + 79T^{2} \) |
| 83 | \( 1 - 4.41T + 83T^{2} \) |
| 89 | \( 1 - 6.51T + 89T^{2} \) |
| 97 | \( 1 - 9.29T + 97T^{2} \) |
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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
−9.381249101696189708644159474558, −8.898337048147081232921117554767, −7.54307222779724472606261714987, −7.10039311426386396245107694910, −6.18477747823944677326506102525, −5.50163418249755768387711277360, −4.01589019617594869824213767059, −3.62596224000353660409307855167, −2.90314974400364331954078124574, −0.888110727848343304722560266180,
0.888110727848343304722560266180, 2.90314974400364331954078124574, 3.62596224000353660409307855167, 4.01589019617594869824213767059, 5.50163418249755768387711277360, 6.18477747823944677326506102525, 7.10039311426386396245107694910, 7.54307222779724472606261714987, 8.898337048147081232921117554767, 9.381249101696189708644159474558