| L(s) = 1 | − 7.84·3-s − 5·5-s + 8.97·7-s + 34.4·9-s + 28.9·11-s + 16.0·13-s + 39.2·15-s + 25.1·17-s + 35.6·19-s − 70.3·21-s + 23·23-s + 25·25-s − 58.7·27-s + 138.·29-s − 40.1·31-s − 226.·33-s − 44.8·35-s + 379.·37-s − 126.·39-s + 412.·41-s + 402.·43-s − 172.·45-s − 110.·47-s − 262.·49-s − 197.·51-s − 421.·53-s − 144.·55-s + ⋯ |
| L(s) = 1 | − 1.50·3-s − 0.447·5-s + 0.484·7-s + 1.27·9-s + 0.792·11-s + 0.343·13-s + 0.674·15-s + 0.359·17-s + 0.430·19-s − 0.731·21-s + 0.208·23-s + 0.200·25-s − 0.418·27-s + 0.885·29-s − 0.232·31-s − 1.19·33-s − 0.216·35-s + 1.68·37-s − 0.517·39-s + 1.57·41-s + 1.42·43-s − 0.571·45-s − 0.341·47-s − 0.765·49-s − 0.542·51-s − 1.09·53-s − 0.354·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.341203970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.341203970\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 7.84T + 27T^{2} \) |
| 7 | \( 1 - 8.97T + 343T^{2} \) |
| 11 | \( 1 - 28.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 35.6T + 6.85e3T^{2} \) |
| 29 | \( 1 - 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 40.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 379.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 412.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 402.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 110.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 421.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 755.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 307.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 319.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 554.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 705.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 455.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.04e3T + 9.12e5T^{2} \) |
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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
−8.972352259557850400267833183725, −7.904998858936953107127214169223, −7.26620112453179681293926569652, −6.22594373556882699030695858011, −5.87872073435787546249854697057, −4.73321201808192667614431346902, −4.29677321442182162091624291967, −3.02335605393003480717260367992, −1.40008039605774596637856636029, −0.66185448991029235996603345230,
0.66185448991029235996603345230, 1.40008039605774596637856636029, 3.02335605393003480717260367992, 4.29677321442182162091624291967, 4.73321201808192667614431346902, 5.87872073435787546249854697057, 6.22594373556882699030695858011, 7.26620112453179681293926569652, 7.904998858936953107127214169223, 8.972352259557850400267833183725
