| L(s) = 1 | − 0.226·2-s + 1.89·3-s − 1.94·4-s − 0.429·6-s − 7-s + 0.894·8-s + 0.590·9-s − 11-s − 3.69·12-s − 2.71·13-s + 0.226·14-s + 3.69·16-s + 6.92·17-s − 0.133·18-s + 6.38·19-s − 1.89·21-s + 0.226·22-s − 2.82·23-s + 1.69·24-s + 0.615·26-s − 4.56·27-s + 1.94·28-s + 4.88·29-s − 7.04·31-s − 2.62·32-s − 1.89·33-s − 1.57·34-s + ⋯ |
| L(s) = 1 | − 0.160·2-s + 1.09·3-s − 0.974·4-s − 0.175·6-s − 0.377·7-s + 0.316·8-s + 0.196·9-s − 0.301·11-s − 1.06·12-s − 0.752·13-s + 0.0605·14-s + 0.923·16-s + 1.68·17-s − 0.0315·18-s + 1.46·19-s − 0.413·21-s + 0.0483·22-s − 0.588·23-s + 0.346·24-s + 0.120·26-s − 0.878·27-s + 0.368·28-s + 0.908·29-s − 1.26·31-s − 0.464·32-s − 0.329·33-s − 0.269·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.712028184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.712028184\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.226T + 2T^{2} \) |
| 3 | \( 1 - 1.89T + 3T^{2} \) |
| 13 | \( 1 + 2.71T + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 - 6.38T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 4.88T + 29T^{2} \) |
| 31 | \( 1 + 7.04T + 31T^{2} \) |
| 37 | \( 1 + 1.27T + 37T^{2} \) |
| 41 | \( 1 - 5.34T + 41T^{2} \) |
| 43 | \( 1 - 9.92T + 43T^{2} \) |
| 47 | \( 1 - 7.64T + 47T^{2} \) |
| 53 | \( 1 - 2.16T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 0.513T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 8.96T + 71T^{2} \) |
| 73 | \( 1 - 6.93T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 9.07T + 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 - 6.05T + 97T^{2} \) |
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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
−9.228875204688079212002052258232, −8.480566766997485210192602577104, −7.68941892085710494965726722582, −7.34907785835550183192979833798, −5.72074272551549025264478276303, −5.25659564844755342477646714934, −3.99235145374420795072922653856, −3.34455918323432408276127780389, −2.44360268667918342811516790917, −0.874107618844005677749972845020,
0.874107618844005677749972845020, 2.44360268667918342811516790917, 3.34455918323432408276127780389, 3.99235145374420795072922653856, 5.25659564844755342477646714934, 5.72074272551549025264478276303, 7.34907785835550183192979833798, 7.68941892085710494965726722582, 8.480566766997485210192602577104, 9.228875204688079212002052258232
