| L(s) = 1 | + 2.41·2-s + 2.82·3-s + 3.82·4-s + 6.82·6-s − 2·7-s + 4.41·8-s + 5.00·9-s + 10.8·12-s − 1.17·13-s − 4.82·14-s + 2.99·16-s + 6.82·17-s + 12.0·18-s − 5.65·21-s + 2.82·23-s + 12.4·24-s − 2.82·26-s + 5.65·27-s − 7.65·28-s + 3.65·29-s − 1.58·32-s + 16.4·34-s + 19.1·36-s + 7.65·37-s − 3.31·39-s − 6·41-s − 13.6·42-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.63·3-s + 1.91·4-s + 2.78·6-s − 0.755·7-s + 1.56·8-s + 1.66·9-s + 3.12·12-s − 0.324·13-s − 1.29·14-s + 0.749·16-s + 1.65·17-s + 2.84·18-s − 1.23·21-s + 0.589·23-s + 2.54·24-s − 0.554·26-s + 1.08·27-s − 1.44·28-s + 0.679·29-s − 0.280·32-s + 2.82·34-s + 3.19·36-s + 1.25·37-s − 0.530·39-s − 0.937·41-s − 2.10·42-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
| \(L(1)\) |
\(\approx\) |
\(8.674340519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.674340519\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 - 2.82T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 1.17T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−8.543595794065615730625616272981, −7.86162874918331236731681670897, −7.09444665063039379876237783115, −6.42934232509989407182661143997, −5.48530395293167692716477900078, −4.65277066717994482718023454751, −3.77809346775746328276417327705, −3.10348990680387451660398411655, −2.78585550565994987651273464998, −1.58298745393235635248050307252,
1.58298745393235635248050307252, 2.78585550565994987651273464998, 3.10348990680387451660398411655, 3.77809346775746328276417327705, 4.65277066717994482718023454751, 5.48530395293167692716477900078, 6.42934232509989407182661143997, 7.09444665063039379876237783115, 7.86162874918331236731681670897, 8.543595794065615730625616272981
