| L(s) = 1 | − 3.16·3-s + 4.24·7-s + 7.00·9-s − 13.4·21-s − 1.41·23-s − 12.6·27-s + 8.94·29-s + 12·41-s + 3.16·43-s + 9.89·47-s + 10.9·49-s + 13.4·61-s + 29.6·63-s − 15.8·67-s + 4.47·69-s + 19.0·81-s − 9.48·83-s − 28.2·87-s − 6·89-s − 8.94·101-s − 12.7·103-s − 9.48·107-s + 13.4·109-s + ⋯ |
| L(s) = 1 | − 1.82·3-s + 1.60·7-s + 2.33·9-s − 2.92·21-s − 0.294·23-s − 2.43·27-s + 1.66·29-s + 1.87·41-s + 0.482·43-s + 1.44·47-s + 1.57·49-s + 1.71·61-s + 3.74·63-s − 1.93·67-s + 0.538·69-s + 2.11·81-s − 1.04·83-s − 3.03·87-s − 0.635·89-s − 0.889·101-s − 1.25·103-s − 0.917·107-s + 1.28·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.391486748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.391486748\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.16T + 3T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 8.94T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 - 9.89T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.84698237854185036091088772531, −7.24126704166503363185719672343, −6.50953763837312410659084863284, −5.70136602064571688255490547342, −5.33788310706385739388702613838, −4.39293310433670662386798343917, −4.25985471766429859919869643486, −2.54250479292172834031722608755, −1.45030015060695382885101537357, −0.75265696496809349060463928416,
0.75265696496809349060463928416, 1.45030015060695382885101537357, 2.54250479292172834031722608755, 4.25985471766429859919869643486, 4.39293310433670662386798343917, 5.33788310706385739388702613838, 5.70136602064571688255490547342, 6.50953763837312410659084863284, 7.24126704166503363185719672343, 7.84698237854185036091088772531
