| L(s) = 1 | + 3-s − 2.82·5-s − 4.24·7-s + 9-s − 4·11-s − 4.24·13-s − 2.82·15-s + 6·17-s − 2·19-s − 4.24·21-s − 2.82·23-s + 3.00·25-s + 27-s + 5.65·29-s + 4.24·31-s − 4·33-s + 12·35-s − 4.24·37-s − 4.24·39-s − 10·41-s + 6·43-s − 2.82·45-s + 2.82·47-s + 10.9·49-s + 6·51-s + 5.65·53-s + 11.3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.26·5-s − 1.60·7-s + 0.333·9-s − 1.20·11-s − 1.17·13-s − 0.730·15-s + 1.45·17-s − 0.458·19-s − 0.925·21-s − 0.589·23-s + 0.600·25-s + 0.192·27-s + 1.05·29-s + 0.762·31-s − 0.696·33-s + 2.02·35-s − 0.697·37-s − 0.679·39-s − 1.56·41-s + 0.914·43-s − 0.421·45-s + 0.412·47-s + 1.57·49-s + 0.840·51-s + 0.777·53-s + 1.52·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7883505339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7883505339\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 - 16T + 73T^{2} \) |
| 79 | \( 1 - 4.24T + 79T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.510699356568066854635759986460, −7.948396338708538476838965701872, −7.33586840954896661299214907324, −6.68511301966452371287594349893, −5.63145989018020156642899237042, −4.71581312533343705134500115381, −3.75524652057486521148577619705, −3.13221744395784393588325089990, −2.44592567828117820293898493449, −0.48941152695802307824438879138,
0.48941152695802307824438879138, 2.44592567828117820293898493449, 3.13221744395784393588325089990, 3.75524652057486521148577619705, 4.71581312533343705134500115381, 5.63145989018020156642899237042, 6.68511301966452371287594349893, 7.33586840954896661299214907324, 7.948396338708538476838965701872, 8.510699356568066854635759986460
