| L(s) = 1 | + 7·5-s + 21·7-s + 6·11-s + 13·13-s + 115·17-s + 46·19-s + 144·23-s − 76·25-s + 162·29-s − 180·31-s + 147·35-s + 13·37-s − 192·41-s + 33·43-s + 383·47-s + 98·49-s − 288·53-s + 42·55-s + 442·59-s − 680·61-s + 91·65-s + 722·67-s − 207·71-s + 274·73-s + 126·77-s + 936·79-s − 1.20e3·83-s + ⋯ |
| L(s) = 1 | + 0.626·5-s + 1.13·7-s + 0.164·11-s + 0.277·13-s + 1.64·17-s + 0.555·19-s + 1.30·23-s − 0.607·25-s + 1.03·29-s − 1.04·31-s + 0.709·35-s + 0.0577·37-s − 0.731·41-s + 0.117·43-s + 1.18·47-s + 2/7·49-s − 0.746·53-s + 0.102·55-s + 0.975·59-s − 1.42·61-s + 0.173·65-s + 1.31·67-s − 0.346·71-s + 0.439·73-s + 0.186·77-s + 1.33·79-s − 1.59·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.557588531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.557588531\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 115 T + p^{3} T^{2} \) |
| 19 | \( 1 - 46 T + p^{3} T^{2} \) |
| 23 | \( 1 - 144 T + p^{3} T^{2} \) |
| 29 | \( 1 - 162 T + p^{3} T^{2} \) |
| 31 | \( 1 + 180 T + p^{3} T^{2} \) |
| 37 | \( 1 - 13 T + p^{3} T^{2} \) |
| 41 | \( 1 + 192 T + p^{3} T^{2} \) |
| 43 | \( 1 - 33 T + p^{3} T^{2} \) |
| 47 | \( 1 - 383 T + p^{3} T^{2} \) |
| 53 | \( 1 + 288 T + p^{3} T^{2} \) |
| 59 | \( 1 - 442 T + p^{3} T^{2} \) |
| 61 | \( 1 + 680 T + p^{3} T^{2} \) |
| 67 | \( 1 - 722 T + p^{3} T^{2} \) |
| 71 | \( 1 + 207 T + p^{3} T^{2} \) |
| 73 | \( 1 - 274 T + p^{3} T^{2} \) |
| 79 | \( 1 - 936 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1204 T + p^{3} T^{2} \) |
| 89 | \( 1 - 966 T + p^{3} T^{2} \) |
| 97 | \( 1 + 138 T + p^{3} T^{2} \) |
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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
−8.901826782215708878460649953225, −8.022095303096988216941686101327, −7.44832115774631302037837075595, −6.44976355644166394511906901157, −5.44649340739693040798376149835, −5.07466339228531419562784624795, −3.87479199434103494647863708531, −2.88498306112293453404444618372, −1.68702901703188181414439591414, −0.974353601692816649912859366023,
0.974353601692816649912859366023, 1.68702901703188181414439591414, 2.88498306112293453404444618372, 3.87479199434103494647863708531, 5.07466339228531419562784624795, 5.44649340739693040798376149835, 6.44976355644166394511906901157, 7.44832115774631302037837075595, 8.022095303096988216941686101327, 8.901826782215708878460649953225
