| L(s) = 1 | + 8·5-s − 7·7-s + 40·11-s − 12·13-s + 58·17-s + 26·19-s + 64·23-s − 61·25-s + 62·29-s + 252·31-s − 56·35-s + 26·37-s − 6·41-s + 416·43-s + 396·47-s + 49·49-s + 450·53-s + 320·55-s − 274·59-s − 576·61-s − 96·65-s − 476·67-s + 448·71-s − 158·73-s − 280·77-s − 936·79-s − 530·83-s + ⋯ |
| L(s) = 1 | + 0.715·5-s − 0.377·7-s + 1.09·11-s − 0.256·13-s + 0.827·17-s + 0.313·19-s + 0.580·23-s − 0.487·25-s + 0.397·29-s + 1.46·31-s − 0.270·35-s + 0.115·37-s − 0.0228·41-s + 1.47·43-s + 1.22·47-s + 1/7·49-s + 1.16·53-s + 0.784·55-s − 0.604·59-s − 1.20·61-s − 0.183·65-s − 0.867·67-s + 0.748·71-s − 0.253·73-s − 0.414·77-s − 1.33·79-s − 0.700·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.120190885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.120190885\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 40 T + p^{3} T^{2} \) |
| 13 | \( 1 + 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 58 T + p^{3} T^{2} \) |
| 19 | \( 1 - 26 T + p^{3} T^{2} \) |
| 23 | \( 1 - 64 T + p^{3} T^{2} \) |
| 29 | \( 1 - 62 T + p^{3} T^{2} \) |
| 31 | \( 1 - 252 T + p^{3} T^{2} \) |
| 37 | \( 1 - 26 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 - 416 T + p^{3} T^{2} \) |
| 47 | \( 1 - 396 T + p^{3} T^{2} \) |
| 53 | \( 1 - 450 T + p^{3} T^{2} \) |
| 59 | \( 1 + 274 T + p^{3} T^{2} \) |
| 61 | \( 1 + 576 T + p^{3} T^{2} \) |
| 67 | \( 1 + 476 T + p^{3} T^{2} \) |
| 71 | \( 1 - 448 T + p^{3} T^{2} \) |
| 73 | \( 1 + 158 T + p^{3} T^{2} \) |
| 79 | \( 1 + 936 T + p^{3} T^{2} \) |
| 83 | \( 1 + 530 T + p^{3} T^{2} \) |
| 89 | \( 1 - 390 T + p^{3} T^{2} \) |
| 97 | \( 1 - 214 T + p^{3} T^{2} \) |
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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
−11.75628537027938603421154214709, −10.48303115972460352880204609081, −9.658014651942685287425535691867, −8.917395854202808495786609625430, −7.57408299517192853968114611141, −6.47447226050115250491706782741, −5.59971072853872372959202044264, −4.18394481633716034358041574267, −2.77266402865700337329642290747, −1.15641280375620641116442613763,
1.15641280375620641116442613763, 2.77266402865700337329642290747, 4.18394481633716034358041574267, 5.59971072853872372959202044264, 6.47447226050115250491706782741, 7.57408299517192853968114611141, 8.917395854202808495786609625430, 9.658014651942685287425535691867, 10.48303115972460352880204609081, 11.75628537027938603421154214709
