| L(s) = 1 | + 2.48·2-s + 4.19·4-s + 2.46·5-s − 3.15·7-s + 5.45·8-s + 6.13·10-s + 0.446·11-s + 2.88·13-s − 7.85·14-s + 5.18·16-s + 1.66·17-s + 8.25·19-s + 10.3·20-s + 1.11·22-s + 1.04·23-s + 1.07·25-s + 7.17·26-s − 13.2·28-s − 1.21·29-s + 5.61·31-s + 2.00·32-s + 4.14·34-s − 7.77·35-s − 10.8·37-s + 20.5·38-s + 13.4·40-s + 41-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 2.09·4-s + 1.10·5-s − 1.19·7-s + 1.92·8-s + 1.93·10-s + 0.134·11-s + 0.799·13-s − 2.09·14-s + 1.29·16-s + 0.403·17-s + 1.89·19-s + 2.31·20-s + 0.236·22-s + 0.218·23-s + 0.215·25-s + 1.40·26-s − 2.49·28-s − 0.225·29-s + 1.00·31-s + 0.354·32-s + 0.710·34-s − 1.31·35-s − 1.77·37-s + 3.33·38-s + 2.12·40-s + 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.577857990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.577857990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 2.48T + 2T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 7 | \( 1 + 3.15T + 7T^{2} \) |
| 11 | \( 1 - 0.446T + 11T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 - 1.66T + 17T^{2} \) |
| 19 | \( 1 - 8.25T + 19T^{2} \) |
| 23 | \( 1 - 1.04T + 23T^{2} \) |
| 29 | \( 1 + 1.21T + 29T^{2} \) |
| 31 | \( 1 - 5.61T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 43 | \( 1 + 6.19T + 43T^{2} \) |
| 47 | \( 1 + 1.17T + 47T^{2} \) |
| 53 | \( 1 - 6.73T + 53T^{2} \) |
| 59 | \( 1 - 15.1T + 59T^{2} \) |
| 61 | \( 1 + 8.39T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 9.95T + 73T^{2} \) |
| 79 | \( 1 - 7.66T + 79T^{2} \) |
| 83 | \( 1 - 2.77T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.686615229157768980579181697197, −7.44815829033172138411174042440, −6.68850753732847748534177234186, −6.21495528679351193446494080332, −5.49154553694892781890851518350, −5.05158700099727993123981301430, −3.72296577621117695163547340691, −3.32166510674258118997797570926, −2.46665903744406525813823357219, −1.32863697384230364561476095337,
1.32863697384230364561476095337, 2.46665903744406525813823357219, 3.32166510674258118997797570926, 3.72296577621117695163547340691, 5.05158700099727993123981301430, 5.49154553694892781890851518350, 6.21495528679351193446494080332, 6.68850753732847748534177234186, 7.44815829033172138411174042440, 8.686615229157768980579181697197
