| L(s) = 1 | − 2.41·3-s + 0.828·7-s + 2.82·9-s + 5.24·11-s + 5.65·13-s + 0.171·17-s + 1.58·19-s − 1.99·21-s − 4.82·23-s + 0.414·27-s + 8·29-s + 0.828·31-s − 12.6·33-s + 7.65·37-s − 13.6·39-s + 10.6·41-s − 10·43-s − 9.65·47-s − 6.31·49-s − 0.414·51-s − 7.65·53-s − 3.82·57-s − 3.65·59-s + 6·61-s + 2.34·63-s + 2.75·67-s + 11.6·69-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 0.313·7-s + 0.942·9-s + 1.58·11-s + 1.56·13-s + 0.0416·17-s + 0.363·19-s − 0.436·21-s − 1.00·23-s + 0.0797·27-s + 1.48·29-s + 0.148·31-s − 2.20·33-s + 1.25·37-s − 2.18·39-s + 1.66·41-s − 1.52·43-s − 1.40·47-s − 0.901·49-s − 0.0580·51-s − 1.05·53-s − 0.507·57-s − 0.476·59-s + 0.768·61-s + 0.295·63-s + 0.336·67-s + 1.40·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.435554745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.435554745\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 11 | \( 1 - 5.24T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 0.171T + 17T^{2} \) |
| 19 | \( 1 - 1.58T + 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 0.828T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 2.75T + 67T^{2} \) |
| 71 | \( 1 - 9.65T + 71T^{2} \) |
| 73 | \( 1 + 5.82T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 1.24T + 83T^{2} \) |
| 89 | \( 1 - 6.17T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−8.581364816939881858823753745376, −8.001400191558416426175734285489, −6.78420721471701699290556995293, −6.27987654973122353872412285551, −5.90312197573458136833185201044, −4.80374413461874990488384192431, −4.19109336870570621566579352390, −3.25499885403743920475537834818, −1.59593792766717613177481207471, −0.858278929847287250913949236585,
0.858278929847287250913949236585, 1.59593792766717613177481207471, 3.25499885403743920475537834818, 4.19109336870570621566579352390, 4.80374413461874990488384192431, 5.90312197573458136833185201044, 6.27987654973122353872412285551, 6.78420721471701699290556995293, 8.001400191558416426175734285489, 8.581364816939881858823753745376
