| L(s) = 1 | + 1.87·2-s + 1.77·3-s + 1.51·4-s + 3.33·6-s − 4.25·7-s − 0.901·8-s + 0.159·9-s + 2.70·12-s − 1.36·13-s − 7.98·14-s − 4.73·16-s − 2.09·17-s + 0.299·18-s + 0.604·19-s − 7.56·21-s + 4.39·23-s − 1.60·24-s − 2.55·26-s − 5.04·27-s − 6.46·28-s − 6.63·29-s − 2.19·31-s − 7.07·32-s − 3.93·34-s + 0.242·36-s − 6.16·37-s + 1.13·38-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 1.02·3-s + 0.759·4-s + 1.36·6-s − 1.60·7-s − 0.318·8-s + 0.0531·9-s + 0.779·12-s − 0.377·13-s − 2.13·14-s − 1.18·16-s − 0.508·17-s + 0.0705·18-s + 0.138·19-s − 1.65·21-s + 0.916·23-s − 0.327·24-s − 0.500·26-s − 0.971·27-s − 1.22·28-s − 1.23·29-s − 0.394·31-s − 1.24·32-s − 0.675·34-s + 0.0403·36-s − 1.01·37-s + 0.183·38-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 | \( 1 - 1.87T + 2T^{2} \) |
| 3 | \( 1 - 1.77T + 3T^{2} \) |
| 7 | \( 1 + 4.25T + 7T^{2} \) |
| 13 | \( 1 + 1.36T + 13T^{2} \) |
| 17 | \( 1 + 2.09T + 17T^{2} \) |
| 19 | \( 1 - 0.604T + 19T^{2} \) |
| 23 | \( 1 - 4.39T + 23T^{2} \) |
| 29 | \( 1 + 6.63T + 29T^{2} \) |
| 31 | \( 1 + 2.19T + 31T^{2} \) |
| 37 | \( 1 + 6.16T + 37T^{2} \) |
| 41 | \( 1 - 7.40T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 3.07T + 47T^{2} \) |
| 53 | \( 1 - 6.65T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 5.68T + 61T^{2} \) |
| 67 | \( 1 + 9.86T + 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 - 0.722T + 73T^{2} \) |
| 79 | \( 1 - 5.67T + 79T^{2} \) |
| 83 | \( 1 - 0.952T + 83T^{2} \) |
| 89 | \( 1 - 1.24T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−8.494923742112327203219155522889, −7.28739229292824355681290345742, −6.80241921090293488879494603066, −5.90481777461327683797876963225, −5.27172735264673565384795411215, −4.16594000799710546674948634642, −3.46193622023666107912745508993, −2.98578651915386719364933636422, −2.17516992994483788831485047740, 0,
2.17516992994483788831485047740, 2.98578651915386719364933636422, 3.46193622023666107912745508993, 4.16594000799710546674948634642, 5.27172735264673565384795411215, 5.90481777461327683797876963225, 6.80241921090293488879494603066, 7.28739229292824355681290345742, 8.494923742112327203219155522889
