| L(s) = 1 | − 9.74·3-s + 20.5·5-s + 29.4·7-s + 68.0·9-s − 61.9·11-s − 11.1·13-s − 200.·15-s − 54.9·17-s + 78.4·19-s − 286.·21-s + 56.6·23-s + 299.·25-s − 399.·27-s − 29·29-s − 151.·31-s + 603.·33-s + 606.·35-s + 272.·37-s + 108.·39-s + 382.·41-s − 29.9·43-s + 1.40e3·45-s + 68.4·47-s + 522.·49-s + 535.·51-s − 108.·53-s − 1.27e3·55-s + ⋯ |
| L(s) = 1 | − 1.87·3-s + 1.84·5-s + 1.58·7-s + 2.51·9-s − 1.69·11-s − 0.238·13-s − 3.45·15-s − 0.783·17-s + 0.947·19-s − 2.98·21-s + 0.513·23-s + 2.39·25-s − 2.85·27-s − 0.185·29-s − 0.875·31-s + 3.18·33-s + 2.92·35-s + 1.20·37-s + 0.447·39-s + 1.45·41-s − 0.106·43-s + 4.64·45-s + 0.212·47-s + 1.52·49-s + 1.47·51-s − 0.281·53-s − 3.12·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.923698239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.923698239\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 9.74T + 27T^{2} \) |
| 5 | \( 1 - 20.5T + 125T^{2} \) |
| 7 | \( 1 - 29.4T + 343T^{2} \) |
| 11 | \( 1 + 61.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 11.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 54.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 78.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 56.6T + 1.21e4T^{2} \) |
| 31 | \( 1 + 151.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 272.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 382.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 29.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 68.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 108.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 617.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 27.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 621.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 225.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 265.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 114.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 700.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 80.4T + 7.04e5T^{2} \) |
| 97 | \( 1 - 549.T + 9.12e5T^{2} \) |
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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
−9.156400713454774777952993390880, −7.83719280596939395051126133576, −7.18752579807200289116585107529, −6.14425791546316090245381334048, −5.55072448917644202257265475822, −5.07909937277862409960186065327, −4.59748631136701935191064902040, −2.47612931332895165033923645918, −1.67701041821813343924201487379, −0.74791970852240104684473045343,
0.74791970852240104684473045343, 1.67701041821813343924201487379, 2.47612931332895165033923645918, 4.59748631136701935191064902040, 5.07909937277862409960186065327, 5.55072448917644202257265475822, 6.14425791546316090245381334048, 7.18752579807200289116585107529, 7.83719280596939395051126133576, 9.156400713454774777952993390880
