| L(s) = 1 | − 2-s − 1.36·3-s + 4-s + 0.286·5-s + 1.36·6-s − 1.35·7-s − 8-s − 1.13·9-s − 0.286·10-s + 0.307·11-s − 1.36·12-s + 3.14·13-s + 1.35·14-s − 0.391·15-s + 16-s + 3.89·17-s + 1.13·18-s + 19-s + 0.286·20-s + 1.84·21-s − 0.307·22-s + 4.29·23-s + 1.36·24-s − 4.91·25-s − 3.14·26-s + 5.64·27-s − 1.35·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.788·3-s + 0.5·4-s + 0.128·5-s + 0.557·6-s − 0.510·7-s − 0.353·8-s − 0.378·9-s − 0.0906·10-s + 0.0928·11-s − 0.394·12-s + 0.872·13-s + 0.361·14-s − 0.101·15-s + 0.250·16-s + 0.944·17-s + 0.267·18-s + 0.229·19-s + 0.0641·20-s + 0.402·21-s − 0.0656·22-s + 0.895·23-s + 0.278·24-s − 0.983·25-s − 0.617·26-s + 1.08·27-s − 0.255·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9898735772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9898735772\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 1.36T + 3T^{2} \) |
| 5 | \( 1 - 0.286T + 5T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 - 0.307T + 11T^{2} \) |
| 13 | \( 1 - 3.14T + 13T^{2} \) |
| 17 | \( 1 - 3.89T + 17T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 - 1.34T + 29T^{2} \) |
| 31 | \( 1 - 8.75T + 31T^{2} \) |
| 37 | \( 1 - 3.45T + 37T^{2} \) |
| 41 | \( 1 - 1.79T + 41T^{2} \) |
| 43 | \( 1 - 6.18T + 43T^{2} \) |
| 47 | \( 1 + 6.83T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6.63T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 3.99T + 67T^{2} \) |
| 71 | \( 1 + 9.68T + 71T^{2} \) |
| 73 | \( 1 + 1.50T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 5.09T + 89T^{2} \) |
| 97 | \( 1 - 3.77T + 97T^{2} \) |
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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
−7.990293506566963026911887393572, −7.05114699819975513268373953110, −6.39580860400931343596024470020, −5.92186460015727987764014168820, −5.30580233383712814984553757470, −4.33433884271317237507566262154, −3.29952055171663634242284771185, −2.71224549397809458519474005996, −1.39589848969765217289743482458, −0.62224114695032850218848986443,
0.62224114695032850218848986443, 1.39589848969765217289743482458, 2.71224549397809458519474005996, 3.29952055171663634242284771185, 4.33433884271317237507566262154, 5.30580233383712814984553757470, 5.92186460015727987764014168820, 6.39580860400931343596024470020, 7.05114699819975513268373953110, 7.990293506566963026911887393572
