L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s + 4·14-s + 16-s + 6·17-s + 4·19-s − 20-s − 6·23-s + 25-s + 4·28-s + 6·29-s + 10·31-s + 32-s + 6·34-s − 4·35-s + 10·37-s + 4·38-s − 40-s − 6·41-s − 4·43-s − 6·46-s + 12·47-s + 9·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s − 1.25·23-s + 1/5·25-s + 0.755·28-s + 1.11·29-s + 1.79·31-s + 0.176·32-s + 1.02·34-s − 0.676·35-s + 1.64·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s − 0.884·46-s + 1.75·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.780265780\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.780265780\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.90013571710671, −15.32765013701608, −14.95068734295928, −14.21014348361752, −13.91550171474862, −13.56241943296718, −12.37914217938914, −12.02912469192993, −11.84424678186356, −11.07217948615906, −10.50545147691836, −9.936468112331188, −9.177792513841780, −8.177518271705935, −7.833356831161605, −7.638349180793635, −6.482027009558117, −5.999514110451615, −5.140314743416681, −4.729881922258882, −4.140204268596983, −3.269739405195133, −2.626310292152584, −1.574489412847685, −0.9382379566034233,
0.9382379566034233, 1.574489412847685, 2.626310292152584, 3.269739405195133, 4.140204268596983, 4.729881922258882, 5.140314743416681, 5.999514110451615, 6.482027009558117, 7.638349180793635, 7.833356831161605, 8.177518271705935, 9.177792513841780, 9.936468112331188, 10.50545147691836, 11.07217948615906, 11.84424678186356, 12.02912469192993, 12.37914217938914, 13.56241943296718, 13.91550171474862, 14.21014348361752, 14.95068734295928, 15.32765013701608, 15.90013571710671