| L(s) = 1 | − 1.67·2-s + 3.10·3-s + 0.795·4-s − 5.19·6-s − 3.08·7-s + 2.01·8-s + 6.67·9-s + 2.47·12-s − 3.37·13-s + 5.16·14-s − 4.95·16-s − 0.103·17-s − 11.1·18-s + 2.03·19-s − 9.60·21-s − 2.23·23-s + 6.26·24-s + 5.64·26-s + 11.4·27-s − 2.45·28-s − 6.49·29-s − 9.07·31-s + 4.26·32-s + 0.173·34-s + 5.30·36-s + 0.333·37-s − 3.39·38-s + ⋯ |
| L(s) = 1 | − 1.18·2-s + 1.79·3-s + 0.397·4-s − 2.12·6-s − 1.16·7-s + 0.711·8-s + 2.22·9-s + 0.714·12-s − 0.937·13-s + 1.38·14-s − 1.23·16-s − 0.0251·17-s − 2.62·18-s + 0.465·19-s − 2.09·21-s − 0.465·23-s + 1.27·24-s + 1.10·26-s + 2.19·27-s − 0.464·28-s − 1.20·29-s − 1.63·31-s + 0.753·32-s + 0.0297·34-s + 0.884·36-s + 0.0548·37-s − 0.550·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 3 | \( 1 - 3.10T + 3T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 + 0.103T + 17T^{2} \) |
| 19 | \( 1 - 2.03T + 19T^{2} \) |
| 23 | \( 1 + 2.23T + 23T^{2} \) |
| 29 | \( 1 + 6.49T + 29T^{2} \) |
| 31 | \( 1 + 9.07T + 31T^{2} \) |
| 37 | \( 1 - 0.333T + 37T^{2} \) |
| 41 | \( 1 - 3.93T + 41T^{2} \) |
| 43 | \( 1 + 7.39T + 43T^{2} \) |
| 47 | \( 1 + 6.62T + 47T^{2} \) |
| 53 | \( 1 + 0.310T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 3.39T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 2.66T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 - 2.41T + 79T^{2} \) |
| 83 | \( 1 + 0.143T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 5.92T + 97T^{2} \) |
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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
−8.504264733520784424988331101579, −7.67516312515927297501781685598, −7.36988758712233118483483137198, −6.55940570956492888778192192999, −5.17534355306151277318620919105, −4.03043342974590396149489373200, −3.37246114961825071825637814641, −2.43568652085518670332473638077, −1.62275125902487993888268717629, 0,
1.62275125902487993888268717629, 2.43568652085518670332473638077, 3.37246114961825071825637814641, 4.03043342974590396149489373200, 5.17534355306151277318620919105, 6.55940570956492888778192192999, 7.36988758712233118483483137198, 7.67516312515927297501781685598, 8.504264733520784424988331101579
