| L(s) = 1 | + 2.51·3-s + 3.24·5-s + 3.64·7-s + 3.32·9-s − 4.55·11-s − 1.90·13-s + 8.17·15-s − 2.76·17-s − 0.000966·19-s + 9.15·21-s + 3.53·23-s + 5.55·25-s + 0.818·27-s + 3.21·29-s + 6.76·31-s − 11.4·33-s + 11.8·35-s − 10.7·37-s − 4.79·39-s + 6.22·41-s − 12.1·43-s + 10.8·45-s − 1.34·47-s + 6.25·49-s − 6.94·51-s − 9.40·53-s − 14.7·55-s + ⋯ |
| L(s) = 1 | + 1.45·3-s + 1.45·5-s + 1.37·7-s + 1.10·9-s − 1.37·11-s − 0.529·13-s + 2.10·15-s − 0.669·17-s − 0.000221·19-s + 1.99·21-s + 0.736·23-s + 1.11·25-s + 0.157·27-s + 0.596·29-s + 1.21·31-s − 1.99·33-s + 1.99·35-s − 1.76·37-s − 0.768·39-s + 0.972·41-s − 1.85·43-s + 1.61·45-s − 0.196·47-s + 0.894·49-s − 0.972·51-s − 1.29·53-s − 1.99·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.651744546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.651744546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 - 2.51T + 3T^{2} \) |
| 5 | \( 1 - 3.24T + 5T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 + 4.55T + 11T^{2} \) |
| 13 | \( 1 + 1.90T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 + 0.000966T + 19T^{2} \) |
| 23 | \( 1 - 3.53T + 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 - 6.76T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 6.22T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 + 1.34T + 47T^{2} \) |
| 53 | \( 1 + 9.40T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 + 2.75T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 5.42T + 71T^{2} \) |
| 73 | \( 1 + 7.26T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 7.89T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−9.504605891679348873590726840619, −8.743183666417258383306198078791, −8.153310420539806906545075815266, −7.46510172336451965880687786082, −6.37966993832852234322413796232, −5.12703304486513147405799104935, −4.75848197246334565900334744726, −3.10878963575651889100080116708, −2.32793836256792444118385325388, −1.67798551894031443946600596847,
1.67798551894031443946600596847, 2.32793836256792444118385325388, 3.10878963575651889100080116708, 4.75848197246334565900334744726, 5.12703304486513147405799104935, 6.37966993832852234322413796232, 7.46510172336451965880687786082, 8.153310420539806906545075815266, 8.743183666417258383306198078791, 9.504605891679348873590726840619
