| L(s) = 1 | + 5.25·3-s + 9.15·7-s + 0.640·9-s + 11.8·11-s + 41.9·13-s + 75.7·17-s − 2.49·19-s + 48.1·21-s − 17.8·23-s − 138.·27-s − 143.·29-s + 88.6·31-s + 62.5·33-s + 351.·37-s + 220.·39-s + 195.·41-s + 366.·43-s + 58.5·47-s − 259.·49-s + 398.·51-s − 0.374·53-s − 13.0·57-s − 318.·59-s + 446.·61-s + 5.86·63-s − 709.·67-s − 93.8·69-s + ⋯ |
| L(s) = 1 | + 1.01·3-s + 0.494·7-s + 0.0237·9-s + 0.326·11-s + 0.895·13-s + 1.08·17-s − 0.0300·19-s + 0.500·21-s − 0.161·23-s − 0.987·27-s − 0.919·29-s + 0.513·31-s + 0.329·33-s + 1.55·37-s + 0.906·39-s + 0.746·41-s + 1.29·43-s + 0.181·47-s − 0.755·49-s + 1.09·51-s − 0.000971·53-s − 0.0304·57-s − 0.703·59-s + 0.937·61-s + 0.0117·63-s − 1.29·67-s − 0.163·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.390930534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.390930534\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 5.25T + 27T^{2} \) |
| 7 | \( 1 - 9.15T + 343T^{2} \) |
| 11 | \( 1 - 11.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 75.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2.49T + 6.85e3T^{2} \) |
| 23 | \( 1 + 17.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 143.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 88.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 351.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 366.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 58.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 0.374T + 1.48e5T^{2} \) |
| 59 | \( 1 + 318.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 446.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 709.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 85.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 926.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 973.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3T + 9.12e5T^{2} \) |
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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
−9.573538361827098111035791706506, −9.076717200312354672452619986590, −8.015922643778242150211022225778, −7.75081666056065701751420116176, −6.35260287842923989285993795479, −5.49080194793796604714803935681, −4.17965308217386089650330220704, −3.35692743213272514983848244914, −2.27312627130631550010688982840, −1.04656360435881183277262263924,
1.04656360435881183277262263924, 2.27312627130631550010688982840, 3.35692743213272514983848244914, 4.17965308217386089650330220704, 5.49080194793796604714803935681, 6.35260287842923989285993795479, 7.75081666056065701751420116176, 8.015922643778242150211022225778, 9.076717200312354672452619986590, 9.573538361827098111035791706506
