| L(s) = 1 | − 7·3-s + 15·5-s + 18·7-s + 22·9-s + 27·11-s + 57·13-s − 105·15-s − 44·17-s + 152·19-s − 126·21-s + 152·23-s + 100·25-s + 35·27-s + 29·29-s + 173·31-s − 189·33-s + 270·35-s + 120·37-s − 399·39-s − 314·41-s + 339·43-s + 330·45-s + 357·47-s − 19·49-s + 308·51-s + 59·53-s + 405·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 1.34·5-s + 0.971·7-s + 0.814·9-s + 0.740·11-s + 1.21·13-s − 1.80·15-s − 0.627·17-s + 1.83·19-s − 1.30·21-s + 1.37·23-s + 4/5·25-s + 0.249·27-s + 0.185·29-s + 1.00·31-s − 0.996·33-s + 1.30·35-s + 0.533·37-s − 1.63·39-s − 1.19·41-s + 1.20·43-s + 1.09·45-s + 1.10·47-s − 0.0553·49-s + 0.845·51-s + 0.152·53-s + 0.992·55-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
| \(L(2)\) |
\(\approx\) |
\(2.766152664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.766152664\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - p T \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 5 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 7 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 27 T + p^{3} T^{2} \) |
| 13 | \( 1 - 57 T + p^{3} T^{2} \) |
| 17 | \( 1 + 44 T + p^{3} T^{2} \) |
| 19 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 31 | \( 1 - 173 T + p^{3} T^{2} \) |
| 37 | \( 1 - 120 T + p^{3} T^{2} \) |
| 41 | \( 1 + 314 T + p^{3} T^{2} \) |
| 43 | \( 1 - 339 T + p^{3} T^{2} \) |
| 47 | \( 1 - 357 T + p^{3} T^{2} \) |
| 53 | \( 1 - 59 T + p^{3} T^{2} \) |
| 59 | \( 1 + 572 T + p^{3} T^{2} \) |
| 61 | \( 1 - 420 T + p^{3} T^{2} \) |
| 67 | \( 1 - 660 T + p^{3} T^{2} \) |
| 71 | \( 1 + 726 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1004 T + p^{3} T^{2} \) |
| 79 | \( 1 + 361 T + p^{3} T^{2} \) |
| 83 | \( 1 + 168 T + p^{3} T^{2} \) |
| 89 | \( 1 - 58 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1206 T + p^{3} T^{2} \) |
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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
−9.068008552631548029082452979437, −8.144834831666827810731721100620, −6.92149044023380334526319004384, −6.40466093302104597123249115488, −5.53821568091147476562957144629, −5.19489715346983136222682352225, −4.19159215555630218806325670029, −2.78508870198901030666091049206, −1.37716615629295503742106142414, −1.02105737664015172072545999875,
1.02105737664015172072545999875, 1.37716615629295503742106142414, 2.78508870198901030666091049206, 4.19159215555630218806325670029, 5.19489715346983136222682352225, 5.53821568091147476562957144629, 6.40466093302104597123249115488, 6.92149044023380334526319004384, 8.144834831666827810731721100620, 9.068008552631548029082452979437
