| L(s) = 1 | + 3.36·5-s + 3.64·7-s + 1.19·11-s + 3.74·13-s − 3.06·17-s + 2.32·19-s − 7.82·23-s + 6.29·25-s − 0.979·29-s + 0.354·31-s + 12.2·35-s − 6.57·37-s − 6.43·41-s + 7.98·43-s + 11.1·47-s + 6.29·49-s − 7.70·53-s + 4·55-s + 11.0·59-s + 6.57·61-s + 12.5·65-s − 5.65·67-s − 3.36·71-s + 7.29·73-s + 4.33·77-s + 4.35·79-s + 1.19·83-s + ⋯ |
| L(s) = 1 | + 1.50·5-s + 1.37·7-s + 0.358·11-s + 1.03·13-s − 0.744·17-s + 0.533·19-s − 1.63·23-s + 1.25·25-s − 0.181·29-s + 0.0636·31-s + 2.07·35-s − 1.08·37-s − 1.00·41-s + 1.21·43-s + 1.63·47-s + 0.898·49-s − 1.05·53-s + 0.539·55-s + 1.43·59-s + 0.841·61-s + 1.55·65-s − 0.691·67-s − 0.399·71-s + 0.853·73-s + 0.494·77-s + 0.489·79-s + 0.130·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.830379329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.830379329\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.36T + 5T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 - 1.19T + 11T^{2} \) |
| 13 | \( 1 - 3.74T + 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 19 | \( 1 - 2.32T + 19T^{2} \) |
| 23 | \( 1 + 7.82T + 23T^{2} \) |
| 29 | \( 1 + 0.979T + 29T^{2} \) |
| 31 | \( 1 - 0.354T + 31T^{2} \) |
| 37 | \( 1 + 6.57T + 37T^{2} \) |
| 41 | \( 1 + 6.43T + 41T^{2} \) |
| 43 | \( 1 - 7.98T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 7.70T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 6.57T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 3.36T + 71T^{2} \) |
| 73 | \( 1 - 7.29T + 73T^{2} \) |
| 79 | \( 1 - 4.35T + 79T^{2} \) |
| 83 | \( 1 - 1.19T + 83T^{2} \) |
| 89 | \( 1 - 9.50T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.78267955038114943252012358280, −6.96011638236127565895956905989, −6.17882903548920012871497733750, −5.72814916219761775528784569059, −5.07613930908748934558678511721, −4.29941529704119130323008192715, −3.50828605302803460815956171904, −2.19401052241475051875409903979, −1.89137154322445915183308966480, −1.00803416753245694295410635162,
1.00803416753245694295410635162, 1.89137154322445915183308966480, 2.19401052241475051875409903979, 3.50828605302803460815956171904, 4.29941529704119130323008192715, 5.07613930908748934558678511721, 5.72814916219761775528784569059, 6.17882903548920012871497733750, 6.96011638236127565895956905989, 7.78267955038114943252012358280
