| L(s) = 1 | + 1.19·2-s − 6.56·4-s − 7·7-s − 17.4·8-s − 4.52·11-s + 14.5·13-s − 8.37·14-s + 31.7·16-s − 15.5·17-s − 14.8·19-s − 5.41·22-s + 166.·23-s + 17.4·26-s + 45.9·28-s + 194.·29-s + 104.·31-s + 177.·32-s − 18.6·34-s − 47.5·37-s − 17.7·38-s − 378.·41-s + 194.·43-s + 29.7·44-s + 199.·46-s − 488.·47-s + 49·49-s − 95.7·52-s + ⋯ |
| L(s) = 1 | + 0.422·2-s − 0.821·4-s − 0.377·7-s − 0.770·8-s − 0.124·11-s + 0.310·13-s − 0.159·14-s + 0.495·16-s − 0.221·17-s − 0.179·19-s − 0.0524·22-s + 1.51·23-s + 0.131·26-s + 0.310·28-s + 1.24·29-s + 0.605·31-s + 0.979·32-s − 0.0938·34-s − 0.211·37-s − 0.0758·38-s − 1.44·41-s + 0.691·43-s + 0.101·44-s + 0.639·46-s − 1.51·47-s + 0.142·49-s − 0.255·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 1.19T + 8T^{2} \) |
| 11 | \( 1 + 4.52T + 1.33e3T^{2} \) |
| 13 | \( 1 - 14.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 14.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 166.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 104.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 47.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 378.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 194.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 316.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 350.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 115.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 434.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 410.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 464.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 290.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 578.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 937.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 839.T + 9.12e5T^{2} \) |
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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
−8.687496907191919963543619272677, −8.054031040373261777762029274148, −6.84737695098697817801058802849, −6.20642830598497974048347734189, −5.14334382571885853409104576655, −4.59197763588451983805407658445, −3.52074950236082126130273708813, −2.79404003475741885108642862281, −1.18601180477300205568132783563, 0,
1.18601180477300205568132783563, 2.79404003475741885108642862281, 3.52074950236082126130273708813, 4.59197763588451983805407658445, 5.14334382571885853409104576655, 6.20642830598497974048347734189, 6.84737695098697817801058802849, 8.054031040373261777762029274148, 8.687496907191919963543619272677
