| L(s) = 1 | + 19.0·3-s + 210.·7-s + 121.·9-s + 18.2·11-s − 834.·13-s + 292.·17-s + 2.28e3·19-s + 4.01e3·21-s + 3.78e3·23-s − 2.31e3·27-s + 4.55e3·29-s + 1.45e3·31-s + 348.·33-s − 5.50e3·37-s − 1.59e4·39-s − 616.·41-s + 1.58e4·43-s − 2.91e4·47-s + 2.73e4·49-s + 5.58e3·51-s + 3.69e4·53-s + 4.36e4·57-s − 1.87e4·59-s + 3.30e4·61-s + 2.55e4·63-s + 5.59e3·67-s + 7.23e4·69-s + ⋯ |
| L(s) = 1 | + 1.22·3-s + 1.62·7-s + 0.500·9-s + 0.0455·11-s − 1.37·13-s + 0.245·17-s + 1.45·19-s + 1.98·21-s + 1.49·23-s − 0.612·27-s + 1.00·29-s + 0.272·31-s + 0.0557·33-s − 0.661·37-s − 1.67·39-s − 0.0573·41-s + 1.30·43-s − 1.92·47-s + 1.62·49-s + 0.300·51-s + 1.80·53-s + 1.78·57-s − 0.699·59-s + 1.13·61-s + 0.810·63-s + 0.152·67-s + 1.82·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.166163975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.166163975\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 19.0T + 243T^{2} \) |
| 7 | \( 1 - 210.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 18.2T + 1.61e5T^{2} \) |
| 13 | \( 1 + 834.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 292.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.28e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.78e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.50e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 616.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.58e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.91e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.69e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.87e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.59e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.45e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.93e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.54e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.33e5T + 8.58e9T^{2} \) |
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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
−10.32784516125422484100096208662, −9.359407779097609189485336117663, −8.572281195328367608289564482522, −7.75148614822102352663357971652, −7.17174831015605830293553280162, −5.32854544813479552140543269258, −4.63447339540823894859140928366, −3.20004830071621181440645059415, −2.27963006745042852886109166019, −1.10138378232870041597595485176,
1.10138378232870041597595485176, 2.27963006745042852886109166019, 3.20004830071621181440645059415, 4.63447339540823894859140928366, 5.32854544813479552140543269258, 7.17174831015605830293553280162, 7.75148614822102352663357971652, 8.572281195328367608289564482522, 9.359407779097609189485336117663, 10.32784516125422484100096208662
