L(s) = 1 | − 1.13·2-s − 1.94·3-s + 0.290·4-s − 5-s + 2.20·6-s + 1.49·7-s + 0.805·8-s + 2.77·9-s + 1.13·10-s − 0.241·11-s − 0.564·12-s + 13-s − 1.70·14-s + 1.94·15-s − 1.20·16-s − 0.709·17-s − 3.14·18-s − 0.290·20-s − 2.90·21-s + 0.273·22-s + 1.77·23-s − 1.56·24-s + 25-s − 1.13·26-s − 3.43·27-s + 0.435·28-s − 2.20·30-s + ⋯ |
L(s) = 1 | − 1.13·2-s − 1.94·3-s + 0.290·4-s − 5-s + 2.20·6-s + 1.49·7-s + 0.805·8-s + 2.77·9-s + 1.13·10-s − 0.241·11-s − 0.564·12-s + 13-s − 1.70·14-s + 1.94·15-s − 1.20·16-s − 0.709·17-s − 3.14·18-s − 0.290·20-s − 2.90·21-s + 0.273·22-s + 1.77·23-s − 1.56·24-s + 25-s − 1.13·26-s − 3.43·27-s + 0.435·28-s − 2.20·30-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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3154855530\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3154855530\) |
\(L(1)\) |
|
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 | \( 1 + 1.13T + T^{2} \) |
| 3 | \( 1 + 1.94T + T^{2} \) |
| 7 | \( 1 - 1.49T + T^{2} \) |
| 11 | \( 1 + 0.241T + T^{2} \) |
| 17 | \( 1 + 0.709T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.77T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.13T + T^{2} \) |
| 47 | \( 1 + 1.77T + T^{2} \) |
| 53 | \( 1 - 0.241T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 0.241T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.49T + T^{2} \) |
| 97 | \( 1 - 1.94T + T^{2} \) |
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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
−9.248726736755246511167251862689, −8.564338434828315120514522737096, −7.67990712680331224039440377427, −7.22291343623956713816771762328, −6.34354510130673875553621904533, −5.10210854754594609853561161621, −4.79343622931051103821214072591, −3.92253042532431160074776311965, −1.64585215054058414052770944440, −0.78841751567386737534281202557,
0.78841751567386737534281202557, 1.64585215054058414052770944440, 3.92253042532431160074776311965, 4.79343622931051103821214072591, 5.10210854754594609853561161621, 6.34354510130673875553621904533, 7.22291343623956713816771762328, 7.67990712680331224039440377427, 8.564338434828315120514522737096, 9.248726736755246511167251862689