| L(s) = 1 | + 0.0628·2-s − 3-s − 1.99·4-s − 0.908·5-s − 0.0628·6-s + 7-s − 0.251·8-s + 9-s − 0.0570·10-s − 0.265·11-s + 1.99·12-s + 1.99·13-s + 0.0628·14-s + 0.908·15-s + 3.97·16-s − 1.51·17-s + 0.0628·18-s − 0.651·19-s + 1.81·20-s − 21-s − 0.0166·22-s + 2.12·23-s + 0.251·24-s − 4.17·25-s + 0.125·26-s − 27-s − 1.99·28-s + ⋯ |
| L(s) = 1 | + 0.0444·2-s − 0.577·3-s − 0.998·4-s − 0.406·5-s − 0.0256·6-s + 0.377·7-s − 0.0888·8-s + 0.333·9-s − 0.0180·10-s − 0.0800·11-s + 0.576·12-s + 0.554·13-s + 0.0168·14-s + 0.234·15-s + 0.994·16-s − 0.368·17-s + 0.0148·18-s − 0.149·19-s + 0.405·20-s − 0.218·21-s − 0.00355·22-s + 0.442·23-s + 0.0512·24-s − 0.835·25-s + 0.0246·26-s − 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0628T + 2T^{2} \) |
| 5 | \( 1 + 0.908T + 5T^{2} \) |
| 11 | \( 1 + 0.265T + 11T^{2} \) |
| 13 | \( 1 - 1.99T + 13T^{2} \) |
| 17 | \( 1 + 1.51T + 17T^{2} \) |
| 19 | \( 1 + 0.651T + 19T^{2} \) |
| 23 | \( 1 - 2.12T + 23T^{2} \) |
| 29 | \( 1 + 6.76T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 - 0.374T + 37T^{2} \) |
| 41 | \( 1 - 8.38T + 41T^{2} \) |
| 43 | \( 1 + 5.86T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 6.85T + 59T^{2} \) |
| 61 | \( 1 + 8.35T + 61T^{2} \) |
| 67 | \( 1 - 9.22T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 7.43T + 89T^{2} \) |
| 97 | \( 1 + 0.720T + 97T^{2} \) |
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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
−7.66509543087246526943234842058, −6.72882685673492804323780220104, −5.98274506870424233366047563492, −5.30445917991692376056035736127, −4.70437999229492939326816907639, −3.97443230147038275825119166653, −3.43251920085309292294564822729, −2.09588664435162490076386019745, −1.01427987417639545912430598893, 0,
1.01427987417639545912430598893, 2.09588664435162490076386019745, 3.43251920085309292294564822729, 3.97443230147038275825119166653, 4.70437999229492939326816907639, 5.30445917991692376056035736127, 5.98274506870424233366047563492, 6.72882685673492804323780220104, 7.66509543087246526943234842058
