| L(s) = 1 | − 2-s + 2.13·3-s + 4-s − 1.04·5-s − 2.13·6-s − 7-s − 8-s + 1.55·9-s + 1.04·10-s − 2.42·11-s + 2.13·12-s + 1.15·13-s + 14-s − 2.23·15-s + 16-s + 3.05·17-s − 1.55·18-s − 1.04·20-s − 2.13·21-s + 2.42·22-s − 0.514·23-s − 2.13·24-s − 3.90·25-s − 1.15·26-s − 3.08·27-s − 28-s + 5.57·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.23·3-s + 0.5·4-s − 0.467·5-s − 0.871·6-s − 0.377·7-s − 0.353·8-s + 0.518·9-s + 0.330·10-s − 0.729·11-s + 0.616·12-s + 0.320·13-s + 0.267·14-s − 0.576·15-s + 0.250·16-s + 0.740·17-s − 0.366·18-s − 0.233·20-s − 0.465·21-s + 0.515·22-s − 0.107·23-s − 0.435·24-s − 0.781·25-s − 0.226·26-s − 0.593·27-s − 0.188·28-s + 1.03·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.13T + 3T^{2} \) |
| 5 | \( 1 + 1.04T + 5T^{2} \) |
| 11 | \( 1 + 2.42T + 11T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 17 | \( 1 - 3.05T + 17T^{2} \) |
| 23 | \( 1 + 0.514T + 23T^{2} \) |
| 29 | \( 1 - 5.57T + 29T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 37 | \( 1 + 1.13T + 37T^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 + 6.37T + 43T^{2} \) |
| 47 | \( 1 + 0.273T + 47T^{2} \) |
| 53 | \( 1 + 6.18T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 9.38T + 61T^{2} \) |
| 67 | \( 1 - 4.12T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 1.05T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 9.44T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−7.956902274617884432351723861461, −7.60905484668647954650309419370, −6.63044616573863033262765929718, −5.88348597960554783097931603277, −4.85249876733555097696046452125, −3.78034972531408382821734034851, −3.14142248137414945968510268977, −2.49657439214453847037448106132, −1.42690565439226072803949189922, 0,
1.42690565439226072803949189922, 2.49657439214453847037448106132, 3.14142248137414945968510268977, 3.78034972531408382821734034851, 4.85249876733555097696046452125, 5.88348597960554783097931603277, 6.63044616573863033262765929718, 7.60905484668647954650309419370, 7.956902274617884432351723861461
