| L(s) = 1 | − 301.·5-s + 1.47e3·7-s + 105.·11-s − 5.77e3·13-s − 9.68e3·17-s + 2.88e4·19-s − 1.03e5·23-s + 1.30e4·25-s − 3.68e4·29-s + 2.58e5·31-s − 4.46e5·35-s + 1.59e5·37-s + 7.74e5·41-s − 4.99e5·43-s + 1.22e6·47-s + 1.35e6·49-s − 1.62e5·53-s − 3.18e4·55-s − 1.43e6·59-s − 2.83e6·61-s + 1.74e6·65-s − 1.90e6·67-s − 5.25e6·71-s + 6.83e5·73-s + 1.55e5·77-s − 6.21e5·79-s − 5.49e6·83-s + ⋯ |
| L(s) = 1 | − 1.08·5-s + 1.62·7-s + 0.0238·11-s − 0.729·13-s − 0.477·17-s + 0.965·19-s − 1.76·23-s + 0.166·25-s − 0.280·29-s + 1.56·31-s − 1.75·35-s + 0.518·37-s + 1.75·41-s − 0.957·43-s + 1.72·47-s + 1.65·49-s − 0.149·53-s − 0.0258·55-s − 0.908·59-s − 1.59·61-s + 0.787·65-s − 0.771·67-s − 1.74·71-s + 0.205·73-s + 0.0388·77-s − 0.141·79-s − 1.05·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 301.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.47e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 105.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.77e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 9.68e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.88e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.03e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.68e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.58e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.59e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.74e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.99e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.22e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.62e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.43e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.83e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.90e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.25e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.83e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.21e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.49e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.35e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.27e6T + 8.07e13T^{2} \) |
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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
−10.03510520961197823049173229684, −8.799893748779948066437244871433, −7.75472241865820424496785952169, −7.58393119090401289040911678238, −5.93410820944490374504353220104, −4.69294917192967288891406647014, −4.12178684445001678351541555725, −2.57276804065428693383591317047, −1.31592709680718044260344516262, 0,
1.31592709680718044260344516262, 2.57276804065428693383591317047, 4.12178684445001678351541555725, 4.69294917192967288891406647014, 5.93410820944490374504353220104, 7.58393119090401289040911678238, 7.75472241865820424496785952169, 8.799893748779948066437244871433, 10.03510520961197823049173229684
