| L(s) = 1 | + 10·3-s + 8·5-s + 73·9-s − 40·11-s + 12·13-s + 80·15-s + 58·17-s − 26·19-s − 64·23-s − 61·25-s + 460·27-s − 62·29-s − 252·31-s − 400·33-s + 26·37-s + 120·39-s − 6·41-s + 416·43-s + 584·45-s + 396·47-s + 580·51-s − 450·53-s − 320·55-s − 260·57-s − 274·59-s + 576·61-s + 96·65-s + ⋯ |
| L(s) = 1 | + 1.92·3-s + 0.715·5-s + 2.70·9-s − 1.09·11-s + 0.256·13-s + 1.37·15-s + 0.827·17-s − 0.313·19-s − 0.580·23-s − 0.487·25-s + 3.27·27-s − 0.397·29-s − 1.46·31-s − 2.11·33-s + 0.115·37-s + 0.492·39-s − 0.0228·41-s + 1.47·43-s + 1.93·45-s + 1.22·47-s + 1.59·51-s − 1.16·53-s − 0.784·55-s − 0.604·57-s − 0.604·59-s + 1.20·61-s + 0.183·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.569119060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.569119060\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 10 T + p^{3} T^{2} \) |
| 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
−12.53453916208687328166388582723, −10.66756051038492536224054053788, −9.820568754070575038488401054642, −9.066656181245569271761296123996, −8.037870393603815297661177626480, −7.33856284106749104272452775141, −5.68224546435678776655103285688, −4.05532482757441270504320121818, −2.83809987263406599459085267299, −1.80133826012550189586077136026,
1.80133826012550189586077136026, 2.83809987263406599459085267299, 4.05532482757441270504320121818, 5.68224546435678776655103285688, 7.33856284106749104272452775141, 8.037870393603815297661177626480, 9.066656181245569271761296123996, 9.820568754070575038488401054642, 10.66756051038492536224054053788, 12.53453916208687328166388582723
