| L(s) = 1 | + 0.138·3-s − 10.0·5-s − 26.9·9-s − 18.6·11-s + 10.0·13-s − 1.39·15-s + 73.6·17-s + 76.0·19-s + 146.·23-s − 23.2·25-s − 7.46·27-s − 157.·29-s + 69.8·31-s − 2.58·33-s + 309.·37-s + 1.39·39-s + 482.·41-s + 17.3·43-s + 272.·45-s − 346.·47-s + 10.1·51-s − 73.4·53-s + 188.·55-s + 10.5·57-s + 704.·59-s + 841.·61-s − 101.·65-s + ⋯ |
| L(s) = 1 | + 0.0265·3-s − 0.902·5-s − 0.999·9-s − 0.511·11-s + 0.215·13-s − 0.0239·15-s + 1.05·17-s + 0.918·19-s + 1.33·23-s − 0.186·25-s − 0.0531·27-s − 1.00·29-s + 0.404·31-s − 0.0136·33-s + 1.37·37-s + 0.00572·39-s + 1.83·41-s + 0.0615·43-s + 0.901·45-s − 1.07·47-s + 0.0279·51-s − 0.190·53-s + 0.461·55-s + 0.0244·57-s + 1.55·59-s + 1.76·61-s − 0.194·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.358262671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.358262671\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.138T + 27T^{2} \) |
| 5 | \( 1 + 10.0T + 125T^{2} \) |
| 11 | \( 1 + 18.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 76.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 146.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 69.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 309.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 482.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 17.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 346.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 73.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 704.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 841.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 891.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 525.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 376.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.23e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 771.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 175.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.31e3T + 9.12e5T^{2} \) |
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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
−11.24143121635533918942014067147, −9.952617823516788725318360717579, −8.981753805551296539776093391556, −7.962239717736107307550122154708, −7.41965080932842015287594919472, −5.97412150101933017076717341729, −5.08046562231734337133129301430, −3.70995855420754359608698849097, −2.76465450362503978854700807388, −0.75861408699973835244799407150,
0.75861408699973835244799407150, 2.76465450362503978854700807388, 3.70995855420754359608698849097, 5.08046562231734337133129301430, 5.97412150101933017076717341729, 7.41965080932842015287594919472, 7.962239717736107307550122154708, 8.981753805551296539776093391556, 9.952617823516788725318360717579, 11.24143121635533918942014067147
