| L(s) = 1 | + 14.2·2-s + 75.7·4-s − 109.·5-s + 411.·7-s − 746.·8-s − 1.56e3·10-s − 1.33e3·11-s − 7.18e3·13-s + 5.87e3·14-s − 2.03e4·16-s + 7.78e3·17-s − 2.79e4·19-s − 8.31e3·20-s − 1.89e4·22-s − 2.34e4·23-s − 6.60e4·25-s − 1.02e5·26-s + 3.11e4·28-s − 7.50e4·29-s − 2.67e4·31-s − 1.94e5·32-s + 1.11e5·34-s − 4.51e4·35-s − 5.65e4·37-s − 3.98e5·38-s + 8.19e4·40-s + 6.10e4·41-s + ⋯ |
| L(s) = 1 | + 1.26·2-s + 0.591·4-s − 0.392·5-s + 0.453·7-s − 0.515·8-s − 0.495·10-s − 0.301·11-s − 0.906·13-s + 0.572·14-s − 1.24·16-s + 0.384·17-s − 0.934·19-s − 0.232·20-s − 0.380·22-s − 0.401·23-s − 0.845·25-s − 1.14·26-s + 0.268·28-s − 0.571·29-s − 0.161·31-s − 1.05·32-s + 0.484·34-s − 0.178·35-s − 0.183·37-s − 1.17·38-s + 0.202·40-s + 0.138·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + 1.33e3T \) |
| good | 2 | \( 1 - 14.2T + 128T^{2} \) |
| 5 | \( 1 + 109.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 411.T + 8.23e5T^{2} \) |
| 13 | \( 1 + 7.18e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 7.78e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.79e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.34e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 7.50e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.67e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.65e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.10e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.45e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.22e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.82e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 6.64e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.01e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.66e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.80e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 9.68e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.25e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.19e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.48e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.20e7T + 8.07e13T^{2} \) |
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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
−12.17439561151823824470153122940, −11.33311995616993186435920134158, −9.948504893835562743175444157525, −8.505215536635437986078463465625, −7.25439835888763591584721461791, −5.82777621622431485975351268471, −4.77222440454773451526632070954, −3.72641336041870515021520595400, −2.25951803507631484938482966305, 0,
2.25951803507631484938482966305, 3.72641336041870515021520595400, 4.77222440454773451526632070954, 5.82777621622431485975351268471, 7.25439835888763591584721461791, 8.505215536635437986078463465625, 9.948504893835562743175444157525, 11.33311995616993186435920134158, 12.17439561151823824470153122940
