L(s) = 1 | − 1.94·2-s + 1.49·3-s + 2.77·4-s + 5-s − 2.90·6-s + 1.13·7-s − 3.43·8-s + 1.24·9-s − 1.94·10-s + 0.709·11-s + 4.14·12-s − 13-s − 2.20·14-s + 1.49·15-s + 3.90·16-s − 1.77·17-s − 2.41·18-s + 2.77·20-s + 1.70·21-s − 1.37·22-s − 0.241·23-s − 5.14·24-s + 25-s + 1.94·26-s + 0.360·27-s + 3.14·28-s − 2.90·30-s + ⋯ |
L(s) = 1 | − 1.94·2-s + 1.49·3-s + 2.77·4-s + 5-s − 2.90·6-s + 1.13·7-s − 3.43·8-s + 1.24·9-s − 1.94·10-s + 0.709·11-s + 4.14·12-s − 13-s − 2.20·14-s + 1.49·15-s + 3.90·16-s − 1.77·17-s − 2.41·18-s + 2.77·20-s + 1.70·21-s − 1.37·22-s − 0.241·23-s − 5.14·24-s + 25-s + 1.94·26-s + 0.360·27-s + 3.14·28-s − 2.90·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.075658375\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.075658375\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + 1.94T + T^{2} \) |
| 3 | \( 1 - 1.49T + T^{2} \) |
| 7 | \( 1 - 1.13T + T^{2} \) |
| 11 | \( 1 - 0.709T + T^{2} \) |
| 17 | \( 1 + 1.77T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 0.241T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.94T + T^{2} \) |
| 47 | \( 1 - 0.241T + T^{2} \) |
| 53 | \( 1 - 0.709T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 0.709T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.13T + T^{2} \) |
| 97 | \( 1 + 1.49T + T^{2} \) |
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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
−9.264748479643630675920452214235, −8.725251287021768837704294158751, −8.107364763971308176035140655065, −7.30492846112170650436488710201, −6.78209452829725625667304858587, −5.65513457445039859392847768437, −4.23955465756395753043709103724, −2.70983948529623984331796936912, −2.17478861680126184669907456665, −1.51671490132805965991072806610,
1.51671490132805965991072806610, 2.17478861680126184669907456665, 2.70983948529623984331796936912, 4.23955465756395753043709103724, 5.65513457445039859392847768437, 6.78209452829725625667304858587, 7.30492846112170650436488710201, 8.107364763971308176035140655065, 8.725251287021768837704294158751, 9.264748479643630675920452214235