L(s) = 1 | − 1.14·3-s + 5-s + 7-s − 1.68·9-s − 11-s + 4.68·13-s − 1.14·15-s − 0.292·17-s − 6.51·19-s − 1.14·21-s − 2.85·23-s + 25-s + 5.37·27-s − 1.43·29-s − 0.978·31-s + 1.14·33-s + 35-s + 0.853·37-s − 5.37·39-s − 6.22·41-s + 10.3·43-s − 1.68·45-s + 9.95·47-s + 49-s + 0.335·51-s + 5.43·53-s − 55-s + ⋯ |
L(s) = 1 | − 0.661·3-s + 0.447·5-s + 0.377·7-s − 0.561·9-s − 0.301·11-s + 1.29·13-s − 0.295·15-s − 0.0709·17-s − 1.49·19-s − 0.250·21-s − 0.595·23-s + 0.200·25-s + 1.03·27-s − 0.267·29-s − 0.175·31-s + 0.199·33-s + 0.169·35-s + 0.140·37-s − 0.860·39-s − 0.972·41-s + 1.57·43-s − 0.251·45-s + 1.45·47-s + 0.142·49-s + 0.0469·51-s + 0.747·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.467893891\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.467893891\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 1.14T + 3T^{2} \) |
| 13 | \( 1 - 4.68T + 13T^{2} \) |
| 17 | \( 1 + 0.292T + 17T^{2} \) |
| 19 | \( 1 + 6.51T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 + 1.43T + 29T^{2} \) |
| 31 | \( 1 + 0.978T + 31T^{2} \) |
| 37 | \( 1 - 0.853T + 37T^{2} \) |
| 41 | \( 1 + 6.22T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 9.95T + 47T^{2} \) |
| 53 | \( 1 - 5.43T + 53T^{2} \) |
| 59 | \( 1 - 9.37T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 0.585T + 67T^{2} \) |
| 71 | \( 1 - 0.335T + 71T^{2} \) |
| 73 | \( 1 + 3.70T + 73T^{2} \) |
| 79 | \( 1 - 2.51T + 79T^{2} \) |
| 83 | \( 1 - 1.70T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 9.10T + 97T^{2} \) |
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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
−8.225409839825281299981704485432, −7.27662010773201902360514801141, −6.38811394097248376949008172520, −5.94028866958454666283743909286, −5.40420467309929948111726515515, −4.46695675645097904113592165854, −3.76825815807959980425604170294, −2.64505691466926336890008562081, −1.82423206261066086749549891403, −0.65214126911301154548990563558,
0.65214126911301154548990563558, 1.82423206261066086749549891403, 2.64505691466926336890008562081, 3.76825815807959980425604170294, 4.46695675645097904113592165854, 5.40420467309929948111726515515, 5.94028866958454666283743909286, 6.38811394097248376949008172520, 7.27662010773201902360514801141, 8.225409839825281299981704485432