| L(s) = 1 | − 3-s + 0.622·5-s − 4.42·7-s + 9-s − 5.80·11-s + 13-s − 0.622·15-s + 2·17-s + 4.42·19-s + 4.42·21-s + 8.85·23-s − 4.61·25-s − 27-s + 2·29-s + 7.18·31-s + 5.80·33-s − 2.75·35-s + 0.755·37-s − 39-s + 3.37·41-s + 7.61·43-s + 0.622·45-s − 1.80·47-s + 12.6·49-s − 2·51-s + 4.75·53-s − 3.61·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.278·5-s − 1.67·7-s + 0.333·9-s − 1.75·11-s + 0.277·13-s − 0.160·15-s + 0.485·17-s + 1.01·19-s + 0.966·21-s + 1.84·23-s − 0.922·25-s − 0.192·27-s + 0.371·29-s + 1.29·31-s + 1.01·33-s − 0.465·35-s + 0.124·37-s − 0.160·39-s + 0.527·41-s + 1.16·43-s + 0.0927·45-s − 0.263·47-s + 1.80·49-s − 0.280·51-s + 0.653·53-s − 0.487·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9751639588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9751639588\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 0.622T + 5T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 + 5.80T + 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4.42T + 19T^{2} \) |
| 23 | \( 1 - 8.85T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 7.18T + 31T^{2} \) |
| 37 | \( 1 - 0.755T + 37T^{2} \) |
| 41 | \( 1 - 3.37T + 41T^{2} \) |
| 43 | \( 1 - 7.61T + 43T^{2} \) |
| 47 | \( 1 + 1.80T + 47T^{2} \) |
| 53 | \( 1 - 4.75T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 8.10T + 61T^{2} \) |
| 67 | \( 1 + 8.04T + 67T^{2} \) |
| 71 | \( 1 - 7.05T + 71T^{2} \) |
| 73 | \( 1 - 7.24T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 3.05T + 83T^{2} \) |
| 89 | \( 1 + 1.86T + 89T^{2} \) |
| 97 | \( 1 + 0.755T + 97T^{2} \) |
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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
−9.777882239138334989602757745499, −9.163006569654549418923599371504, −7.896180746913405507447611433303, −7.18890579725124698173710425059, −6.23711812070079829379982267602, −5.62421876259807927385982633257, −4.75870903709063069451579861391, −3.32251210244946787787970140738, −2.67591872543313035699312406857, −0.73211253131823077536198210954,
0.73211253131823077536198210954, 2.67591872543313035699312406857, 3.32251210244946787787970140738, 4.75870903709063069451579861391, 5.62421876259807927385982633257, 6.23711812070079829379982267602, 7.18890579725124698173710425059, 7.896180746913405507447611433303, 9.163006569654549418923599371504, 9.777882239138334989602757745499
