| L(s) = 1 | − 0.0650·2-s + 3-s − 1.99·4-s − 2.65·5-s − 0.0650·6-s + 0.259·8-s + 9-s + 0.172·10-s − 5.47·11-s − 1.99·12-s − 3.40·13-s − 2.65·15-s + 3.97·16-s − 6.28·17-s − 0.0650·18-s − 4.25·19-s + 5.29·20-s + 0.355·22-s − 1.06·23-s + 0.259·24-s + 2.04·25-s + 0.221·26-s + 27-s − 6.10·29-s + 0.172·30-s − 10.4·31-s − 0.778·32-s + ⋯ |
| L(s) = 1 | − 0.0459·2-s + 0.577·3-s − 0.997·4-s − 1.18·5-s − 0.0265·6-s + 0.0918·8-s + 0.333·9-s + 0.0545·10-s − 1.64·11-s − 0.576·12-s − 0.943·13-s − 0.685·15-s + 0.993·16-s − 1.52·17-s − 0.0153·18-s − 0.976·19-s + 1.18·20-s + 0.0758·22-s − 0.222·23-s + 0.0530·24-s + 0.408·25-s + 0.0434·26-s + 0.192·27-s − 1.13·29-s + 0.0315·30-s − 1.88·31-s − 0.137·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09002644102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09002644102\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0650T + 2T^{2} \) |
| 5 | \( 1 + 2.65T + 5T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 13 | \( 1 + 3.40T + 13T^{2} \) |
| 17 | \( 1 + 6.28T + 17T^{2} \) |
| 19 | \( 1 + 4.25T + 19T^{2} \) |
| 23 | \( 1 + 1.06T + 23T^{2} \) |
| 29 | \( 1 + 6.10T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 3.83T + 37T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 8.14T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 + 6.51T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 4.82T + 73T^{2} \) |
| 79 | \( 1 - 9.32T + 79T^{2} \) |
| 83 | \( 1 + 5.03T + 83T^{2} \) |
| 89 | \( 1 - 8.62T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−8.017832525710565062887427639758, −7.57857603340988308308673793325, −7.04243475929185777177001104126, −5.72502213598185844988730145490, −5.04106201069290687366665039954, −4.24380050754456911015716743216, −3.89996661121039202513279287386, −2.80792191626232064803855460518, −2.03705992713979251600879412711, −0.14427006576761520870222172578,
0.14427006576761520870222172578, 2.03705992713979251600879412711, 2.80792191626232064803855460518, 3.89996661121039202513279287386, 4.24380050754456911015716743216, 5.04106201069290687366665039954, 5.72502213598185844988730145490, 7.04243475929185777177001104126, 7.57857603340988308308673793325, 8.017832525710565062887427639758
