| L(s) = 1 | + 0.0830·2-s + 0.315·3-s − 1.99·4-s + 2.25·5-s + 0.0262·6-s − 3.20·7-s − 0.331·8-s − 2.90·9-s + 0.186·10-s − 0.218·11-s − 0.629·12-s − 4.17·13-s − 0.266·14-s + 0.710·15-s + 3.95·16-s − 4.68·17-s − 0.240·18-s + 3.43·19-s − 4.48·20-s − 1.01·21-s − 0.0181·22-s − 3.99·23-s − 0.104·24-s + 0.0635·25-s − 0.347·26-s − 1.86·27-s + 6.39·28-s + ⋯ |
| L(s) = 1 | + 0.0587·2-s + 0.182·3-s − 0.996·4-s + 1.00·5-s + 0.0107·6-s − 1.21·7-s − 0.117·8-s − 0.966·9-s + 0.0590·10-s − 0.0657·11-s − 0.181·12-s − 1.15·13-s − 0.0711·14-s + 0.183·15-s + 0.989·16-s − 1.13·17-s − 0.0567·18-s + 0.789·19-s − 1.00·20-s − 0.220·21-s − 0.00386·22-s − 0.833·23-s − 0.0213·24-s + 0.0127·25-s − 0.0680·26-s − 0.358·27-s + 1.20·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 503 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0830T + 2T^{2} \) |
| 3 | \( 1 - 0.315T + 3T^{2} \) |
| 5 | \( 1 - 2.25T + 5T^{2} \) |
| 7 | \( 1 + 3.20T + 7T^{2} \) |
| 11 | \( 1 + 0.218T + 11T^{2} \) |
| 13 | \( 1 + 4.17T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 19 | \( 1 - 3.43T + 19T^{2} \) |
| 23 | \( 1 + 3.99T + 23T^{2} \) |
| 29 | \( 1 + 0.712T + 29T^{2} \) |
| 31 | \( 1 - 1.04T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 41 | \( 1 - 3.18T + 41T^{2} \) |
| 43 | \( 1 + 6.35T + 43T^{2} \) |
| 47 | \( 1 + 3.87T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 2.63T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 6.09T + 67T^{2} \) |
| 71 | \( 1 - 9.40T + 71T^{2} \) |
| 73 | \( 1 + 2.78T + 73T^{2} \) |
| 79 | \( 1 - 5.16T + 79T^{2} \) |
| 83 | \( 1 - 1.83T + 83T^{2} \) |
| 89 | \( 1 + 8.08T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−9.956274510837681971206349104133, −9.700735021215489979572089006310, −8.951367158355819332615274656248, −7.931231294352630285822202401773, −6.58118395239002092219668518263, −5.76097042951575866711337732612, −4.85735929312955023060199259222, −3.47317555755971725644831080250, −2.38032565242628041145185462801, 0,
2.38032565242628041145185462801, 3.47317555755971725644831080250, 4.85735929312955023060199259222, 5.76097042951575866711337732612, 6.58118395239002092219668518263, 7.931231294352630285822202401773, 8.951367158355819332615274656248, 9.700735021215489979572089006310, 9.956274510837681971206349104133
