L(s) = 1 | + 5.54·2-s + 22.7·4-s − 17.7·7-s + 81.4·8-s − 11·11-s + 53.5·13-s − 98.2·14-s + 269.·16-s − 112.·17-s − 1.24·19-s − 60.9·22-s + 78.4·23-s + 296.·26-s − 402.·28-s + 174.·29-s + 82.5·31-s + 842.·32-s − 622.·34-s + 149.·37-s − 6.89·38-s + 414.·41-s + 182.·43-s − 249.·44-s + 434.·46-s + 438.·47-s − 28.5·49-s + 1.21e3·52-s + ⋯ |
L(s) = 1 | + 1.95·2-s + 2.83·4-s − 0.957·7-s + 3.60·8-s − 0.301·11-s + 1.14·13-s − 1.87·14-s + 4.21·16-s − 1.60·17-s − 0.0150·19-s − 0.590·22-s + 0.710·23-s + 2.23·26-s − 2.71·28-s + 1.11·29-s + 0.478·31-s + 4.65·32-s − 3.14·34-s + 0.662·37-s − 0.0294·38-s + 1.57·41-s + 0.647·43-s − 0.855·44-s + 1.39·46-s + 1.36·47-s − 0.0833·49-s + 3.24·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(9.055731387\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.055731387\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
good | 2 | \( 1 - 5.54T + 8T^{2} \) |
| 7 | \( 1 + 17.7T + 343T^{2} \) |
| 13 | \( 1 - 53.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 112.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 1.24T + 6.85e3T^{2} \) |
| 23 | \( 1 - 78.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 82.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 182.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 438.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 490.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 6.02T + 2.05e5T^{2} \) |
| 61 | \( 1 - 434.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 935.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 510.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.04e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 226.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 825.T + 9.12e5T^{2} \) |
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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
−8.422932280526224686736417388936, −7.36685376359466594804795539578, −6.62740273805025938033039290519, −6.19222564393882483692647236157, −5.45331660493585234174699420418, −4.40735672365152337784015130397, −3.96779362548733993050201171160, −2.90829954985456003327152661939, −2.41603670004645563111512850544, −1.00608632849520257233458555231,
1.00608632849520257233458555231, 2.41603670004645563111512850544, 2.90829954985456003327152661939, 3.96779362548733993050201171160, 4.40735672365152337784015130397, 5.45331660493585234174699420418, 6.19222564393882483692647236157, 6.62740273805025938033039290519, 7.36685376359466594804795539578, 8.422932280526224686736417388936