| L(s) = 1 | + 2·2-s + 1.56·3-s + 4·4-s + 3.13·6-s − 3.11·7-s + 8·8-s − 24.5·9-s − 7.96·11-s + 6.26·12-s + 86.3·13-s − 6.23·14-s + 16·16-s + 129.·17-s − 49.0·18-s − 100.·19-s − 4.88·21-s − 15.9·22-s − 23·23-s + 12.5·24-s + 172.·26-s − 80.7·27-s − 12.4·28-s − 152.·29-s + 285.·31-s + 32·32-s − 12.4·33-s + 259.·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.301·3-s + 0.5·4-s + 0.213·6-s − 0.168·7-s + 0.353·8-s − 0.909·9-s − 0.218·11-s + 0.150·12-s + 1.84·13-s − 0.119·14-s + 0.250·16-s + 1.84·17-s − 0.642·18-s − 1.21·19-s − 0.0507·21-s − 0.154·22-s − 0.208·23-s + 0.106·24-s + 1.30·26-s − 0.575·27-s − 0.0841·28-s − 0.978·29-s + 1.65·31-s + 0.176·32-s − 0.0658·33-s + 1.30·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.866280314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.866280314\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 1.56T + 27T^{2} \) |
| 7 | \( 1 + 3.11T + 343T^{2} \) |
| 11 | \( 1 + 7.96T + 1.33e3T^{2} \) |
| 13 | \( 1 - 86.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 129.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 100.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 152.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 285.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 20.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 143.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 125.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 174.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 393.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 824.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 17.4T + 2.26e5T^{2} \) |
| 67 | \( 1 - 753.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 687.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 139.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 764.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 502.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 239.T + 9.12e5T^{2} \) |
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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
−9.420941329592737950819603395673, −8.288583599145024048022669392162, −8.057619854792712418400852622270, −6.65363241304050251399263351699, −5.96213351437570461842890453361, −5.29948218619626668682071351425, −3.92966152499196094579057050381, −3.36682205755721306109176675974, −2.28738746327316521368913399876, −0.924902988749888045554505596483,
0.924902988749888045554505596483, 2.28738746327316521368913399876, 3.36682205755721306109176675974, 3.92966152499196094579057050381, 5.29948218619626668682071351425, 5.96213351437570461842890453361, 6.65363241304050251399263351699, 8.057619854792712418400852622270, 8.288583599145024048022669392162, 9.420941329592737950819603395673
