| L(s) = 1 | − 3.40·3-s + 2.99·5-s − 3.46·7-s + 8.62·9-s + 13-s − 10.1·15-s + 6.00·17-s + 4.10·19-s + 11.8·21-s + 5.25·23-s + 3.94·25-s − 19.1·27-s + 1.94·29-s − 2.23·31-s − 10.3·35-s + 7.55·37-s − 3.40·39-s − 3.29·41-s + 8.24·43-s + 25.7·45-s + 6.40·47-s + 4.99·49-s − 20.4·51-s + 6.62·53-s − 13.9·57-s − 4.62·59-s − 4.97·61-s + ⋯ |
| L(s) = 1 | − 1.96·3-s + 1.33·5-s − 1.30·7-s + 2.87·9-s + 0.277·13-s − 2.63·15-s + 1.45·17-s + 0.941·19-s + 2.57·21-s + 1.09·23-s + 0.789·25-s − 3.68·27-s + 0.360·29-s − 0.401·31-s − 1.75·35-s + 1.24·37-s − 0.545·39-s − 0.515·41-s + 1.25·43-s + 3.84·45-s + 0.934·47-s + 0.713·49-s − 2.86·51-s + 0.910·53-s − 1.85·57-s − 0.602·59-s − 0.636·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.317101150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.317101150\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 3.40T + 3T^{2} \) |
| 5 | \( 1 - 2.99T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 17 | \( 1 - 6.00T + 17T^{2} \) |
| 19 | \( 1 - 4.10T + 19T^{2} \) |
| 23 | \( 1 - 5.25T + 23T^{2} \) |
| 29 | \( 1 - 1.94T + 29T^{2} \) |
| 31 | \( 1 + 2.23T + 31T^{2} \) |
| 37 | \( 1 - 7.55T + 37T^{2} \) |
| 41 | \( 1 + 3.29T + 41T^{2} \) |
| 43 | \( 1 - 8.24T + 43T^{2} \) |
| 47 | \( 1 - 6.40T + 47T^{2} \) |
| 53 | \( 1 - 6.62T + 53T^{2} \) |
| 59 | \( 1 + 4.62T + 59T^{2} \) |
| 61 | \( 1 + 4.97T + 61T^{2} \) |
| 67 | \( 1 - 1.08T + 67T^{2} \) |
| 71 | \( 1 - 2.79T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 - 0.381T + 79T^{2} \) |
| 83 | \( 1 - 3.52T + 83T^{2} \) |
| 89 | \( 1 - 4.38T + 89T^{2} \) |
| 97 | \( 1 - 5.92T + 97T^{2} \) |
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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
−7.64871693753052842385184638047, −7.04738537417293595701336829604, −6.38304954180986871700154323009, −5.77822955696665005366371013250, −5.58745136074414685252235579068, −4.75416891903839236896631409889, −3.71596558559870545052690613831, −2.75089766509387209090441666988, −1.39071254975360620449752664283, −0.74614171418304314641481611461,
0.74614171418304314641481611461, 1.39071254975360620449752664283, 2.75089766509387209090441666988, 3.71596558559870545052690613831, 4.75416891903839236896631409889, 5.58745136074414685252235579068, 5.77822955696665005366371013250, 6.38304954180986871700154323009, 7.04738537417293595701336829604, 7.64871693753052842385184638047
