| L(s) = 1 | + 0.601·3-s + 1.88·5-s − 2.63·9-s − 1.80·11-s − 2.51·13-s + 1.13·15-s − 0.931·17-s − 0.907·19-s + 3.02·23-s − 1.43·25-s − 3.39·27-s + 2.27·29-s + 9.40·31-s − 1.08·33-s − 0.460·37-s − 1.51·39-s − 2.12·41-s − 5.41·43-s − 4.98·45-s + 4.39·47-s − 0.559·51-s − 11.0·53-s − 3.40·55-s − 0.545·57-s + 8.53·59-s + 0.951·61-s − 4.75·65-s + ⋯ |
| L(s) = 1 | + 0.347·3-s + 0.844·5-s − 0.879·9-s − 0.543·11-s − 0.698·13-s + 0.293·15-s − 0.225·17-s − 0.208·19-s + 0.630·23-s − 0.287·25-s − 0.652·27-s + 0.423·29-s + 1.68·31-s − 0.188·33-s − 0.0757·37-s − 0.242·39-s − 0.332·41-s − 0.825·43-s − 0.742·45-s + 0.640·47-s − 0.0783·51-s − 1.51·53-s − 0.458·55-s − 0.0722·57-s + 1.11·59-s + 0.121·61-s − 0.589·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.601T + 3T^{2} \) |
| 5 | \( 1 - 1.88T + 5T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 + 2.51T + 13T^{2} \) |
| 17 | \( 1 + 0.931T + 17T^{2} \) |
| 19 | \( 1 + 0.907T + 19T^{2} \) |
| 23 | \( 1 - 3.02T + 23T^{2} \) |
| 29 | \( 1 - 2.27T + 29T^{2} \) |
| 31 | \( 1 - 9.40T + 31T^{2} \) |
| 37 | \( 1 + 0.460T + 37T^{2} \) |
| 41 | \( 1 + 2.12T + 41T^{2} \) |
| 43 | \( 1 + 5.41T + 43T^{2} \) |
| 47 | \( 1 - 4.39T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 8.53T + 59T^{2} \) |
| 61 | \( 1 - 0.951T + 61T^{2} \) |
| 67 | \( 1 + 3.73T + 67T^{2} \) |
| 71 | \( 1 + 0.271T + 71T^{2} \) |
| 73 | \( 1 + 6.63T + 73T^{2} \) |
| 79 | \( 1 + 8.23T + 79T^{2} \) |
| 83 | \( 1 + 8.96T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−7.997759703175819465395779285729, −7.02901652473820224665279336514, −6.36389714062532837994376477112, −5.59033008225590972664310845162, −5.02830573871953848119297634421, −4.13485183354663945470952735756, −2.86047527550165093623372838264, −2.61561038442908930369024686884, −1.49680842320956724077629930955, 0,
1.49680842320956724077629930955, 2.61561038442908930369024686884, 2.86047527550165093623372838264, 4.13485183354663945470952735756, 5.02830573871953848119297634421, 5.59033008225590972664310845162, 6.36389714062532837994376477112, 7.02901652473820224665279336514, 7.997759703175819465395779285729
