L(s) = 1 | + (−0.5 + 0.866i)7-s + (−1 + 1.73i)13-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)25-s − 31-s + (−1 + 1.73i)37-s + (0.5 + 0.866i)43-s + (−0.499 − 0.866i)49-s − 61-s + 2·67-s + (0.5 + 0.866i)73-s + 2·79-s + (−0.999 − 1.73i)91-s + (0.5 + 0.866i)97-s + (−1 − 1.73i)103-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)7-s + (−1 + 1.73i)13-s + (0.5 − 0.866i)19-s + (−0.5 + 0.866i)25-s − 31-s + (−1 + 1.73i)37-s + (0.5 + 0.866i)43-s + (−0.499 − 0.866i)49-s − 61-s + 2·67-s + (0.5 + 0.866i)73-s + 2·79-s + (−0.999 − 1.73i)91-s + (0.5 + 0.866i)97-s + (−1 − 1.73i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8232431775\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8232431775\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.388152575347637883517687112718, −8.908253215641278667539417307322, −7.87012152210215705770148037674, −6.95282322104477309159354470456, −6.50257354702596145581126262931, −5.38417474670869195490618224536, −4.77685309444115870662052277471, −3.67709149001288713055609250933, −2.66977540780856329821354567858, −1.76171378308491608254147435682,
0.54158261353856604625540368453, 2.16374952905319969606544242252, 3.30775238875518169290618052046, 3.94483216189817510174041837858, 5.14793573013088712172720722882, 5.73992924070072981030773410551, 6.75521607995530625804622122707, 7.62974951133162645292250236582, 7.905706890169707268367623280753, 9.106506408278561392699201247540