| L(s) = 1 | + 8.96·3-s + 5·5-s − 7·7-s + 53.4·9-s − 16.1·11-s − 33.2·13-s + 44.8·15-s − 15.0·17-s − 58.6·19-s − 62.7·21-s − 144.·23-s + 25·25-s + 236.·27-s − 31.7·29-s − 264.·31-s − 144.·33-s − 35·35-s − 3.15·37-s − 297.·39-s + 14.6·41-s + 90.7·43-s + 267.·45-s − 572.·47-s + 49·49-s − 134.·51-s − 343.·53-s − 80.7·55-s + ⋯ |
| L(s) = 1 | + 1.72·3-s + 0.447·5-s − 0.377·7-s + 1.97·9-s − 0.442·11-s − 0.708·13-s + 0.771·15-s − 0.214·17-s − 0.707·19-s − 0.652·21-s − 1.31·23-s + 0.200·25-s + 1.68·27-s − 0.203·29-s − 1.53·31-s − 0.763·33-s − 0.169·35-s − 0.0140·37-s − 1.22·39-s + 0.0558·41-s + 0.321·43-s + 0.884·45-s − 1.77·47-s + 0.142·49-s − 0.369·51-s − 0.891·53-s − 0.197·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 + 7T \) |
| good | 3 | \( 1 - 8.96T + 27T^{2} \) |
| 11 | \( 1 + 16.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 58.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 144.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 31.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 264.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 3.15T + 5.06e4T^{2} \) |
| 41 | \( 1 - 14.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 90.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 572.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 343.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 303.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 380.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 68.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 513.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 921.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 909.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 76.1T + 9.12e5T^{2} \) |
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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
−8.299515154500770753191096943109, −7.72150940176188750833634882642, −6.96391532988237444254323496028, −6.05299499174192957726734698057, −4.93529989272381450297621514358, −3.98635635651844416133312841246, −3.21440223751314517951380850247, −2.31367742334370712603835567473, −1.77048976209437716351269478350, 0,
1.77048976209437716351269478350, 2.31367742334370712603835567473, 3.21440223751314517951380850247, 3.98635635651844416133312841246, 4.93529989272381450297621514358, 6.05299499174192957726734698057, 6.96391532988237444254323496028, 7.72150940176188750833634882642, 8.299515154500770753191096943109
