L(s) = 1 | + 2·2-s − 9.93·3-s + 4·4-s + 8.55·5-s − 19.8·6-s + 30.9·7-s + 8·8-s + 71.6·9-s + 17.1·10-s − 46.2·11-s − 39.7·12-s + 64.5·13-s + 61.9·14-s − 84.9·15-s + 16·16-s − 10.3·17-s + 143.·18-s + 14.7·19-s + 34.2·20-s − 307.·21-s − 92.4·22-s + 33.8·23-s − 79.4·24-s − 51.8·25-s + 129.·26-s − 443.·27-s + 123.·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.91·3-s + 0.5·4-s + 0.765·5-s − 1.35·6-s + 1.67·7-s + 0.353·8-s + 2.65·9-s + 0.541·10-s − 1.26·11-s − 0.955·12-s + 1.37·13-s + 1.18·14-s − 1.46·15-s + 0.250·16-s − 0.148·17-s + 1.87·18-s + 0.177·19-s + 0.382·20-s − 3.19·21-s − 0.895·22-s + 0.307·23-s − 0.675·24-s − 0.414·25-s + 0.973·26-s − 3.15·27-s + 0.835·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.634562235\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.634562235\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 37 | \( 1 - 37T \) |
good | 3 | \( 1 + 9.93T + 27T^{2} \) |
| 5 | \( 1 - 8.55T + 125T^{2} \) |
| 7 | \( 1 - 30.9T + 343T^{2} \) |
| 11 | \( 1 + 46.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 14.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 33.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 69.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 162.T + 2.97e4T^{2} \) |
| 41 | \( 1 - 112.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 444.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 223.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 713.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 138.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 253.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 136.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 272.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 335.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 239.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 336.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 597.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 701.T + 9.12e5T^{2} \) |
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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
−13.78503783440370369667242557648, −12.95251357849662631844549682288, −11.67209198641366183556688952362, −11.04519888684332984169263915437, −10.24778440034677909657849119491, −7.936441830182450231785779155032, −6.37176639925715880742506295631, −5.41351776109249466082860247921, −4.65862504702484858646454520761, −1.49522337671883843159128814768,
1.49522337671883843159128814768, 4.65862504702484858646454520761, 5.41351776109249466082860247921, 6.37176639925715880742506295631, 7.936441830182450231785779155032, 10.24778440034677909657849119491, 11.04519888684332984169263915437, 11.67209198641366183556688952362, 12.95251357849662631844549682288, 13.78503783440370369667242557648