L(s) = 1 | + 3.22·3-s − 3.58·5-s + 7-s + 7.37·9-s − 11-s − 13-s − 11.5·15-s − 4.82·17-s − 7.88·19-s + 3.22·21-s + 4.80·23-s + 7.87·25-s + 14.0·27-s + 9.62·29-s − 5.44·31-s − 3.22·33-s − 3.58·35-s + 0.187·37-s − 3.22·39-s + 1.50·41-s − 9.53·43-s − 26.4·45-s − 3.00·47-s + 49-s − 15.5·51-s − 2.34·53-s + 3.58·55-s + ⋯ |
L(s) = 1 | + 1.85·3-s − 1.60·5-s + 0.377·7-s + 2.45·9-s − 0.301·11-s − 0.277·13-s − 2.98·15-s − 1.16·17-s − 1.80·19-s + 0.702·21-s + 1.00·23-s + 1.57·25-s + 2.70·27-s + 1.78·29-s − 0.978·31-s − 0.560·33-s − 0.606·35-s + 0.0309·37-s − 0.515·39-s + 0.235·41-s − 1.45·43-s − 3.94·45-s − 0.438·47-s + 0.142·49-s − 2.17·51-s − 0.322·53-s + 0.483·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 3.22T + 3T^{2} \) |
| 5 | \( 1 + 3.58T + 5T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 + 7.88T + 19T^{2} \) |
| 23 | \( 1 - 4.80T + 23T^{2} \) |
| 29 | \( 1 - 9.62T + 29T^{2} \) |
| 31 | \( 1 + 5.44T + 31T^{2} \) |
| 37 | \( 1 - 0.187T + 37T^{2} \) |
| 41 | \( 1 - 1.50T + 41T^{2} \) |
| 43 | \( 1 + 9.53T + 43T^{2} \) |
| 47 | \( 1 + 3.00T + 47T^{2} \) |
| 53 | \( 1 + 2.34T + 53T^{2} \) |
| 59 | \( 1 + 4.59T + 59T^{2} \) |
| 61 | \( 1 - 5.05T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 9.44T + 89T^{2} \) |
| 97 | \( 1 + 0.672T + 97T^{2} \) |
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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
−7.72951509789919819134626771411, −6.99674982404976231110721390413, −6.59054727292721150241187020206, −4.89173211323678543935472747554, −4.41028910123918844220623856012, −3.90277540842764211455230270263, −3.04347141026417679497029286901, −2.49383204906576317536015207055, −1.51140627665835562867639780584, 0,
1.51140627665835562867639780584, 2.49383204906576317536015207055, 3.04347141026417679497029286901, 3.90277540842764211455230270263, 4.41028910123918844220623856012, 4.89173211323678543935472747554, 6.59054727292721150241187020206, 6.99674982404976231110721390413, 7.72951509789919819134626771411