| L(s) = 1 | + 2-s − 0.806·3-s + 4-s + 5-s − 0.806·6-s + 3.35·7-s + 8-s − 2.35·9-s + 10-s + 3.76·11-s − 0.806·12-s + 2.80·13-s + 3.35·14-s − 0.806·15-s + 16-s − 3.35·17-s − 2.35·18-s − 4.31·19-s + 20-s − 2.70·21-s + 3.76·22-s − 2.96·23-s − 0.806·24-s + 25-s + 2.80·26-s + 4.31·27-s + 3.35·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.465·3-s + 0.5·4-s + 0.447·5-s − 0.329·6-s + 1.26·7-s + 0.353·8-s − 0.783·9-s + 0.316·10-s + 1.13·11-s − 0.232·12-s + 0.778·13-s + 0.895·14-s − 0.208·15-s + 0.250·16-s − 0.812·17-s − 0.553·18-s − 0.989·19-s + 0.223·20-s − 0.589·21-s + 0.803·22-s − 0.617·23-s − 0.164·24-s + 0.200·25-s + 0.550·26-s + 0.829·27-s + 0.633·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.747559077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.747559077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + 0.806T + 3T^{2} \) |
| 7 | \( 1 - 3.35T + 7T^{2} \) |
| 11 | \( 1 - 3.76T + 11T^{2} \) |
| 13 | \( 1 - 2.80T + 13T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 + 4.31T + 19T^{2} \) |
| 23 | \( 1 + 2.96T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 37 | \( 1 - 9.50T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 9.89T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 - 6.38T + 59T^{2} \) |
| 61 | \( 1 - 2.23T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 9.92T + 71T^{2} \) |
| 73 | \( 1 + 3.66T + 73T^{2} \) |
| 79 | \( 1 - 0.836T + 79T^{2} \) |
| 83 | \( 1 - 3.50T + 83T^{2} \) |
| 89 | \( 1 - 0.700T + 89T^{2} \) |
| 97 | \( 1 + 0.261T + 97T^{2} \) |
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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
−7.56214294797396885228353161567, −6.77048391724320089428038743034, −6.12359844407092522185842510973, −5.68402130746476528739352929859, −5.01592476748687956900935302678, −4.11150344743603411285596815954, −3.81653820056869084393550526666, −2.40372040971154992956142397753, −1.92308993338408618808627260183, −0.868796913384187607581816268855,
0.868796913384187607581816268855, 1.92308993338408618808627260183, 2.40372040971154992956142397753, 3.81653820056869084393550526666, 4.11150344743603411285596815954, 5.01592476748687956900935302678, 5.68402130746476528739352929859, 6.12359844407092522185842510973, 6.77048391724320089428038743034, 7.56214294797396885228353161567
