| L(s) = 1 | + 4·2-s + 16·4-s + 39.9·5-s − 85.8·7-s + 64·8-s + 159.·10-s + 240.·11-s − 343.·14-s + 256·16-s + 639.·20-s + 962.·22-s + 972.·25-s − 1.37e3·28-s + 818·29-s − 1.37e3·31-s + 1.02e3·32-s − 3.43e3·35-s + 2.55e3·40-s + 3.85e3·44-s + 4.96e3·49-s + 3.89e3·50-s + 2.37e3·53-s + 9.62e3·55-s − 5.49e3·56-s + 3.27e3·58-s − 6.86e3·59-s − 5.49e3·62-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 1.59·5-s − 1.75·7-s + 8-s + 1.59·10-s + 1.98·11-s − 1.75·14-s + 16-s + 1.59·20-s + 1.98·22-s + 1.55·25-s − 1.75·28-s + 0.972·29-s − 1.42·31-s + 32-s − 2.80·35-s + 1.59·40-s + 1.98·44-s + 2.06·49-s + 1.55·50-s + 0.846·53-s + 3.18·55-s − 1.75·56-s + 0.972·58-s − 1.97·59-s − 1.42·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(4.548382521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.548382521\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 39.9T + 625T^{2} \) |
| 7 | \( 1 + 85.8T + 2.40e3T^{2} \) |
| 11 | \( 1 - 240.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 2.85e4T^{2} \) |
| 17 | \( 1 - 8.35e4T^{2} \) |
| 19 | \( 1 - 1.30e5T^{2} \) |
| 23 | \( 1 - 2.79e5T^{2} \) |
| 29 | \( 1 - 818T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.37e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.41e6T^{2} \) |
| 47 | \( 1 - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.37e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 6.86e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.86e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 9.11e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 9.28e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.50e4T + 8.85e7T^{2} \) |
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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
−12.02951783867068545325985688941, −10.65236177543510878759151035612, −9.690692064818591743031039294994, −9.109228008659545326262349403439, −6.87707233220923188959162915445, −6.39721403535345579166425181559, −5.62422536360054807850068993410, −3.99147529087032663907428208666, −2.86332146375591210772586427906, −1.47020803365910127492534811995,
1.47020803365910127492534811995, 2.86332146375591210772586427906, 3.99147529087032663907428208666, 5.62422536360054807850068993410, 6.39721403535345579166425181559, 6.87707233220923188959162915445, 9.109228008659545326262349403439, 9.690692064818591743031039294994, 10.65236177543510878759151035612, 12.02951783867068545325985688941
