| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 2.24·11-s + 12-s − 5.82·13-s + 16-s − 6.65·17-s − 18-s + 5.41·19-s − 2.24·22-s − 4.41·23-s − 24-s + 5.82·26-s + 27-s + 0.656·29-s − 1.24·31-s − 32-s + 2.24·33-s + 6.65·34-s + 36-s + 8.24·37-s − 5.41·38-s − 5.82·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.676·11-s + 0.288·12-s − 1.61·13-s + 0.250·16-s − 1.61·17-s − 0.235·18-s + 1.24·19-s − 0.478·22-s − 0.920·23-s − 0.204·24-s + 1.14·26-s + 0.192·27-s + 0.121·29-s − 0.223·31-s − 0.176·32-s + 0.390·33-s + 1.14·34-s + 0.166·36-s + 1.35·37-s − 0.878·38-s − 0.933·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 + 5.82T + 13T^{2} \) |
| 17 | \( 1 + 6.65T + 17T^{2} \) |
| 19 | \( 1 - 5.41T + 19T^{2} \) |
| 23 | \( 1 + 4.41T + 23T^{2} \) |
| 29 | \( 1 - 0.656T + 29T^{2} \) |
| 31 | \( 1 + 1.24T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 - 6.41T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 8.65T + 53T^{2} \) |
| 59 | \( 1 + 4.41T + 59T^{2} \) |
| 61 | \( 1 + 8.65T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 0.242T + 73T^{2} \) |
| 79 | \( 1 + 2.24T + 79T^{2} \) |
| 83 | \( 1 - 8.41T + 83T^{2} \) |
| 89 | \( 1 + 7.31T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−7.66920806050289540110001828022, −7.04179204885918823875940091589, −6.42572435023661323931638582910, −5.52516475090098826695516691950, −4.55898128570346078399449768362, −3.97183861648782052606208895118, −2.77593223191264811479013311001, −2.33966041802478290014560773669, −1.29401060343727134137564740177, 0,
1.29401060343727134137564740177, 2.33966041802478290014560773669, 2.77593223191264811479013311001, 3.97183861648782052606208895118, 4.55898128570346078399449768362, 5.52516475090098826695516691950, 6.42572435023661323931638582910, 7.04179204885918823875940091589, 7.66920806050289540110001828022
