| L(s) = 1 | + 0.414·3-s − 4.41·7-s − 2.82·9-s + 1.41·11-s + 5.82·13-s + 17-s + 19-s − 1.82·21-s − 0.757·23-s − 2.41·27-s − 0.171·29-s − 6.24·31-s + 0.585·33-s − 8.48·37-s + 2.41·39-s − 4.24·41-s − 1.75·43-s + 12.4·49-s + 0.414·51-s + 5.48·53-s + 0.414·57-s − 6.89·59-s + 14.2·61-s + 12.4·63-s − 4.75·67-s − 0.313·69-s + 13.4·71-s + ⋯ |
| L(s) = 1 | + 0.239·3-s − 1.66·7-s − 0.942·9-s + 0.426·11-s + 1.61·13-s + 0.242·17-s + 0.229·19-s − 0.398·21-s − 0.157·23-s − 0.464·27-s − 0.0318·29-s − 1.12·31-s + 0.101·33-s − 1.39·37-s + 0.386·39-s − 0.662·41-s − 0.267·43-s + 1.78·49-s + 0.0580·51-s + 0.753·53-s + 0.0548·57-s − 0.898·59-s + 1.82·61-s + 1.57·63-s − 0.581·67-s − 0.0377·69-s + 1.59·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.393770425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.393770425\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 5.82T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 23 | \( 1 + 0.757T + 23T^{2} \) |
| 29 | \( 1 + 0.171T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 1.75T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 5.48T + 53T^{2} \) |
| 59 | \( 1 + 6.89T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 + 4.75T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 6.48T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + 0.343T + 97T^{2} \) |
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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
−8.057334228531402187935844275678, −6.89591318699788309308585209639, −6.60925611297014590718413892820, −5.78064395466502431290077905884, −5.36874951342787396190054201957, −3.88724104573993141454737070634, −3.55261147468150429076037296787, −2.93655084899245770171479323750, −1.82375861032489316295646955709, −0.56987093657706897724803345373,
0.56987093657706897724803345373, 1.82375861032489316295646955709, 2.93655084899245770171479323750, 3.55261147468150429076037296787, 3.88724104573993141454737070634, 5.36874951342787396190054201957, 5.78064395466502431290077905884, 6.60925611297014590718413892820, 6.89591318699788309308585209639, 8.057334228531402187935844275678
