| L(s) = 1 | + 3-s − 5-s − 0.828·7-s + 9-s − 4.82·11-s − 2·13-s − 15-s − 5.65·17-s − 1.17·19-s − 0.828·21-s + 6.82·23-s + 25-s + 27-s + 6·29-s + 4·31-s − 4.82·33-s + 0.828·35-s − 3.65·37-s − 2·39-s + 7.65·41-s + 1.65·43-s − 45-s + 10.8·47-s − 6.31·49-s − 5.65·51-s + 13.3·53-s + 4.82·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.313·7-s + 0.333·9-s − 1.45·11-s − 0.554·13-s − 0.258·15-s − 1.37·17-s − 0.268·19-s − 0.180·21-s + 1.42·23-s + 0.200·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.840·33-s + 0.140·35-s − 0.601·37-s − 0.320·39-s + 1.19·41-s + 0.252·43-s − 0.149·45-s + 1.57·47-s − 0.901·49-s − 0.792·51-s + 1.82·53-s + 0.651·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.582783822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.582783822\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 2.48T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.466281665337689554999685172513, −7.81750997955453101160926808680, −7.08302751521369008680522892424, −6.50263622406099209673595091136, −5.29846986803573329753187940041, −4.69976905520934099194864110916, −3.85479650163365616221820829952, −2.71734635423501771002740596717, −2.41292000118030871232243765753, −0.67790247892366330650954601368,
0.67790247892366330650954601368, 2.41292000118030871232243765753, 2.71734635423501771002740596717, 3.85479650163365616221820829952, 4.69976905520934099194864110916, 5.29846986803573329753187940041, 6.50263622406099209673595091136, 7.08302751521369008680522892424, 7.81750997955453101160926808680, 8.466281665337689554999685172513
