| L(s) = 1 | + 9.61·2-s − 35.5·4-s − 125·5-s − 1.12e3·7-s − 1.57e3·8-s − 1.20e3·10-s + 7.27e3·11-s − 6.31e3·13-s − 1.08e4·14-s − 1.05e4·16-s − 3.63e4·17-s + 5.24e3·19-s + 4.43e3·20-s + 6.99e4·22-s − 7.76e4·23-s + 1.56e4·25-s − 6.07e4·26-s + 4.00e4·28-s + 1.91e4·29-s − 2.60e5·31-s + 9.95e4·32-s − 3.49e5·34-s + 1.41e5·35-s + 1.92e5·37-s + 5.04e4·38-s + 1.96e5·40-s − 3.71e5·41-s + ⋯ |
| L(s) = 1 | + 0.850·2-s − 0.277·4-s − 0.447·5-s − 1.24·7-s − 1.08·8-s − 0.380·10-s + 1.64·11-s − 0.797·13-s − 1.05·14-s − 0.645·16-s − 1.79·17-s + 0.175·19-s + 0.124·20-s + 1.40·22-s − 1.33·23-s + 0.199·25-s − 0.678·26-s + 0.345·28-s + 0.145·29-s − 1.57·31-s + 0.537·32-s − 1.52·34-s + 0.556·35-s + 0.624·37-s + 0.149·38-s + 0.485·40-s − 0.840·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.9519808378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9519808378\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| good | 2 | \( 1 - 9.61T + 128T^{2} \) |
| 7 | \( 1 + 1.12e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.27e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.31e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.63e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.24e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.76e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.91e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.60e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.92e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.71e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.28e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.70e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.36e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 3.06e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.38e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.87e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.89e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.05e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.84e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.98e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.95e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.37e6T + 8.07e13T^{2} \) |
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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
−9.798949993211985486916502205156, −9.318063657278061405913411250000, −8.383929155119126817585793383838, −6.77625186964025913821965155680, −6.45229577906464414648520522988, −5.13560179728954234722248716714, −4.01049075774423781916149435882, −3.58379524217494926145433421985, −2.20516278182008437798078088708, −0.37389161737796636981504478828,
0.37389161737796636981504478828, 2.20516278182008437798078088708, 3.58379524217494926145433421985, 4.01049075774423781916149435882, 5.13560179728954234722248716714, 6.45229577906464414648520522988, 6.77625186964025913821965155680, 8.383929155119126817585793383838, 9.318063657278061405913411250000, 9.798949993211985486916502205156
