L(s) = 1 | − 0.178·2-s − 1.96·4-s + 2.75·5-s + 3.42·7-s + 0.707·8-s − 0.490·10-s − 11-s + 13-s − 0.611·14-s + 3.81·16-s + 2.82·17-s − 1.24·19-s − 5.41·20-s + 0.178·22-s + 3.70·23-s + 2.57·25-s − 0.178·26-s − 6.74·28-s − 2.18·29-s − 0.285·31-s − 2.09·32-s − 0.503·34-s + 9.43·35-s − 6.16·37-s + 0.221·38-s + 1.94·40-s + 5.67·41-s + ⋯ |
L(s) = 1 | − 0.126·2-s − 0.984·4-s + 1.23·5-s + 1.29·7-s + 0.250·8-s − 0.155·10-s − 0.301·11-s + 0.277·13-s − 0.163·14-s + 0.952·16-s + 0.684·17-s − 0.285·19-s − 1.21·20-s + 0.0380·22-s + 0.772·23-s + 0.514·25-s − 0.0349·26-s − 1.27·28-s − 0.404·29-s − 0.0513·31-s − 0.370·32-s − 0.0863·34-s + 1.59·35-s − 1.01·37-s + 0.0359·38-s + 0.307·40-s + 0.886·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.848135243\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.848135243\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 0.178T + 2T^{2} \) |
| 5 | \( 1 - 2.75T + 5T^{2} \) |
| 7 | \( 1 - 3.42T + 7T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 - 3.70T + 23T^{2} \) |
| 29 | \( 1 + 2.18T + 29T^{2} \) |
| 31 | \( 1 + 0.285T + 31T^{2} \) |
| 37 | \( 1 + 6.16T + 37T^{2} \) |
| 41 | \( 1 - 5.67T + 41T^{2} \) |
| 43 | \( 1 + 1.42T + 43T^{2} \) |
| 47 | \( 1 - 2.68T + 47T^{2} \) |
| 53 | \( 1 - 7.75T + 53T^{2} \) |
| 59 | \( 1 + 4.12T + 59T^{2} \) |
| 61 | \( 1 + 3.85T + 61T^{2} \) |
| 67 | \( 1 + 8.93T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 - 7.00T + 73T^{2} \) |
| 79 | \( 1 - 2.26T + 79T^{2} \) |
| 83 | \( 1 - 18.1T + 83T^{2} \) |
| 89 | \( 1 - 8.71T + 89T^{2} \) |
| 97 | \( 1 - 0.384T + 97T^{2} \) |
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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
−9.544365436762793103434944387301, −8.939796859584226573214192529208, −8.174765839204499282023254425750, −7.41932276638608354069881232779, −6.11215590174570274432819771209, −5.30597803357904821874068296311, −4.81591829739708304183417136809, −3.62531301753966563141677945084, −2.16622770836686093860526138631, −1.13613332692143935640802598006,
1.13613332692143935640802598006, 2.16622770836686093860526138631, 3.62531301753966563141677945084, 4.81591829739708304183417136809, 5.30597803357904821874068296311, 6.11215590174570274432819771209, 7.41932276638608354069881232779, 8.174765839204499282023254425750, 8.939796859584226573214192529208, 9.544365436762793103434944387301