| L(s) = 1 | + 2.75·2-s − 3-s + 5.60·4-s + 5-s − 2.75·6-s + 1.66·7-s + 9.94·8-s + 9-s + 2.75·10-s − 5.60·12-s − 3.84·13-s + 4.60·14-s − 15-s + 16.2·16-s + 1.66·17-s + 2.75·18-s + 5.51·19-s + 5.60·20-s − 1.66·21-s − 9.21·23-s − 9.94·24-s + 25-s − 10.6·26-s − 27-s + 9.36·28-s − 8.85·29-s − 2.75·30-s + ⋯ |
| L(s) = 1 | + 1.95·2-s − 0.577·3-s + 2.80·4-s + 0.447·5-s − 1.12·6-s + 0.631·7-s + 3.51·8-s + 0.333·9-s + 0.872·10-s − 1.61·12-s − 1.06·13-s + 1.23·14-s − 0.258·15-s + 4.05·16-s + 0.405·17-s + 0.650·18-s + 1.26·19-s + 1.25·20-s − 0.364·21-s − 1.92·23-s − 2.02·24-s + 0.200·25-s − 2.07·26-s − 0.192·27-s + 1.76·28-s − 1.64·29-s − 0.503·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.900494605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.900494605\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.75T + 2T^{2} \) |
| 7 | \( 1 - 1.66T + 7T^{2} \) |
| 13 | \( 1 + 3.84T + 13T^{2} \) |
| 17 | \( 1 - 1.66T + 17T^{2} \) |
| 19 | \( 1 - 5.51T + 19T^{2} \) |
| 23 | \( 1 + 9.21T + 23T^{2} \) |
| 29 | \( 1 + 8.85T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 + 9.36T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 - 3.33T + 61T^{2} \) |
| 67 | \( 1 - 1.21T + 67T^{2} \) |
| 71 | \( 1 + 9.21T + 71T^{2} \) |
| 73 | \( 1 + 7.18T + 73T^{2} \) |
| 79 | \( 1 - 5.51T + 79T^{2} \) |
| 83 | \( 1 - 0.505T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 4.78T + 97T^{2} \) |
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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
−9.723043191305143762320445228317, −7.912625639572664617631286859770, −7.45909827024922482178013478271, −6.50130593037571381401084912465, −5.73765480492050181214723266192, −5.22050649993004836452360529245, −4.50450983148574077854431798202, −3.60324089075882518100245231495, −2.47519976775313353024965436545, −1.58103459079575489535843938290,
1.58103459079575489535843938290, 2.47519976775313353024965436545, 3.60324089075882518100245231495, 4.50450983148574077854431798202, 5.22050649993004836452360529245, 5.73765480492050181214723266192, 6.50130593037571381401084912465, 7.45909827024922482178013478271, 7.912625639572664617631286859770, 9.723043191305143762320445228317
