| L(s) = 1 | + 2.23·3-s + 2·5-s + 2.23·7-s + 2.00·9-s − 2.23·11-s + 4·13-s + 4.47·15-s + 2·17-s + 5.00·21-s + 4.47·23-s − 25-s − 2.23·27-s − 5.00·33-s + 4.47·35-s − 37-s + 8.94·39-s − 5·41-s + 4.00·45-s − 6.70·47-s − 1.99·49-s + 4.47·51-s + 11·53-s − 4.47·55-s + 4.47·59-s + 10·61-s + 4.47·63-s + 8·65-s + ⋯ |
| L(s) = 1 | + 1.29·3-s + 0.894·5-s + 0.845·7-s + 0.666·9-s − 0.674·11-s + 1.10·13-s + 1.15·15-s + 0.485·17-s + 1.09·21-s + 0.932·23-s − 0.200·25-s − 0.430·27-s − 0.870·33-s + 0.755·35-s − 0.164·37-s + 1.43·39-s − 0.780·41-s + 0.596·45-s − 0.978·47-s − 0.285·49-s + 0.626·51-s + 1.51·53-s − 0.603·55-s + 0.582·59-s + 1.28·61-s + 0.563·63-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.759512811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.759512811\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 6.70T + 47T^{2} \) |
| 53 | \( 1 - 11T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 2.23T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.770915516225511129656756493296, −8.373601353218368240650901733076, −7.69277741261152725721168313151, −6.77780822410576543743267623304, −5.71892968424005870197664637906, −5.10746613083261804971143009275, −3.93414056019174397460316934676, −3.07519096189325692528851203511, −2.19504183806685956732857271106, −1.35987706249098983631648901552,
1.35987706249098983631648901552, 2.19504183806685956732857271106, 3.07519096189325692528851203511, 3.93414056019174397460316934676, 5.10746613083261804971143009275, 5.71892968424005870197664637906, 6.77780822410576543743267623304, 7.69277741261152725721168313151, 8.373601353218368240650901733076, 8.770915516225511129656756493296
