| L(s) = 1 | + 1.92·2-s + 1.97·3-s + 1.71·4-s + 3.81·6-s + 3.74·7-s − 0.553·8-s + 0.916·9-s + 0.698·11-s + 3.38·12-s + 4.67·13-s + 7.22·14-s − 4.49·16-s + 5.65·17-s + 1.76·18-s + 0.797·19-s + 7.41·21-s + 1.34·22-s − 2.87·23-s − 1.09·24-s + 9.00·26-s − 4.12·27-s + 6.42·28-s + 0.568·29-s − 5.19·31-s − 7.54·32-s + 1.38·33-s + 10.8·34-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 1.14·3-s + 0.856·4-s + 1.55·6-s + 1.41·7-s − 0.195·8-s + 0.305·9-s + 0.210·11-s + 0.978·12-s + 1.29·13-s + 1.93·14-s − 1.12·16-s + 1.37·17-s + 0.416·18-s + 0.182·19-s + 1.61·21-s + 0.286·22-s − 0.599·23-s − 0.223·24-s + 1.76·26-s − 0.793·27-s + 1.21·28-s + 0.105·29-s − 0.933·31-s − 1.33·32-s + 0.240·33-s + 1.86·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.734457412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.734457412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 1.92T + 2T^{2} \) |
| 3 | \( 1 - 1.97T + 3T^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 - 0.698T + 11T^{2} \) |
| 13 | \( 1 - 4.67T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 - 0.797T + 19T^{2} \) |
| 23 | \( 1 + 2.87T + 23T^{2} \) |
| 29 | \( 1 - 0.568T + 29T^{2} \) |
| 31 | \( 1 + 5.19T + 31T^{2} \) |
| 37 | \( 1 - 5.93T + 37T^{2} \) |
| 41 | \( 1 - 4.54T + 41T^{2} \) |
| 43 | \( 1 + 7.00T + 43T^{2} \) |
| 47 | \( 1 + 3.53T + 47T^{2} \) |
| 53 | \( 1 - 3.89T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 0.157T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 6.17T + 71T^{2} \) |
| 73 | \( 1 + 4.72T + 73T^{2} \) |
| 79 | \( 1 + 4.93T + 79T^{2} \) |
| 83 | \( 1 + 1.99T + 83T^{2} \) |
| 89 | \( 1 - 6.08T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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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
−8.100215794235806005552265216384, −7.49421493378458933984655879808, −6.48196425484105130248746903237, −5.61498461393623389541006706822, −5.25345063001445417404765854464, −4.16286914186305238893204264216, −3.78083235955206671259704514678, −3.03977976517062747010953867839, −2.15364597976179565221176229112, −1.27678222482974721279578153773,
1.27678222482974721279578153773, 2.15364597976179565221176229112, 3.03977976517062747010953867839, 3.78083235955206671259704514678, 4.16286914186305238893204264216, 5.25345063001445417404765854464, 5.61498461393623389541006706822, 6.48196425484105130248746903237, 7.49421493378458933984655879808, 8.100215794235806005552265216384
