| L(s) = 1 | − 3.24·3-s + 5-s − 1.24·7-s + 7.52·9-s + 6.19·11-s + 1.66·13-s − 3.24·15-s + 5.66·17-s + 4.94·19-s + 4.03·21-s + 4.48·23-s + 25-s − 14.6·27-s − 4.48·29-s − 0.385·31-s − 20.0·33-s − 1.24·35-s + 37-s − 5.40·39-s − 0.190·41-s − 4.56·43-s + 7.52·45-s + 6.08·47-s − 5.45·49-s − 18.3·51-s + 1.70·53-s + 6.19·55-s + ⋯ |
| L(s) = 1 | − 1.87·3-s + 0.447·5-s − 0.470·7-s + 2.50·9-s + 1.86·11-s + 0.462·13-s − 0.837·15-s + 1.37·17-s + 1.13·19-s + 0.880·21-s + 0.935·23-s + 0.200·25-s − 2.82·27-s − 0.833·29-s − 0.0692·31-s − 3.49·33-s − 0.210·35-s + 0.164·37-s − 0.865·39-s − 0.0297·41-s − 0.695·43-s + 1.12·45-s + 0.887·47-s − 0.778·49-s − 2.57·51-s + 0.233·53-s + 0.834·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.332194273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.332194273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 3.24T + 3T^{2} \) |
| 7 | \( 1 + 1.24T + 7T^{2} \) |
| 11 | \( 1 - 6.19T + 11T^{2} \) |
| 13 | \( 1 - 1.66T + 13T^{2} \) |
| 17 | \( 1 - 5.66T + 17T^{2} \) |
| 19 | \( 1 - 4.94T + 19T^{2} \) |
| 23 | \( 1 - 4.48T + 23T^{2} \) |
| 29 | \( 1 + 4.48T + 29T^{2} \) |
| 31 | \( 1 + 0.385T + 31T^{2} \) |
| 41 | \( 1 + 0.190T + 41T^{2} \) |
| 43 | \( 1 + 4.56T + 43T^{2} \) |
| 47 | \( 1 - 6.08T + 47T^{2} \) |
| 53 | \( 1 - 1.70T + 53T^{2} \) |
| 59 | \( 1 - 6.10T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 + 0.786T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 5.33T + 89T^{2} \) |
| 97 | \( 1 + 0.561T + 97T^{2} \) |
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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
−9.109957026257617122379060941406, −7.69979012671644691104563741833, −6.84666469882966131754947861140, −6.46711072132189849821310118289, −5.64419739588587604891787412058, −5.22583676194416762711093747779, −4.11100564886281731391978783997, −3.37380411819933157047651251243, −1.48532616280680045334356300557, −0.897787802618397102339712087190,
0.897787802618397102339712087190, 1.48532616280680045334356300557, 3.37380411819933157047651251243, 4.11100564886281731391978783997, 5.22583676194416762711093747779, 5.64419739588587604891787412058, 6.46711072132189849821310118289, 6.84666469882966131754947861140, 7.69979012671644691104563741833, 9.109957026257617122379060941406
