| L(s) = 1 | − 1.95·3-s + 2.85·5-s + 2.57·7-s + 0.823·9-s − 1.95·11-s + 2.85·13-s − 5.59·15-s + 7.03·17-s − 4.97·19-s − 5.03·21-s + 8.60·23-s + 3.17·25-s + 4.25·27-s − 29-s − 4.53·31-s + 3.82·33-s + 7.36·35-s − 7.71·37-s − 5.59·39-s + 6.68·41-s + 3.19·43-s + 2.35·45-s − 3.29·47-s − 0.365·49-s − 13.7·51-s + 3.21·53-s − 5.59·55-s + ⋯ |
| L(s) = 1 | − 1.12·3-s + 1.27·5-s + 0.973·7-s + 0.274·9-s − 0.589·11-s + 0.793·13-s − 1.44·15-s + 1.70·17-s − 1.14·19-s − 1.09·21-s + 1.79·23-s + 0.635·25-s + 0.819·27-s − 0.185·29-s − 0.813·31-s + 0.665·33-s + 1.24·35-s − 1.26·37-s − 0.895·39-s + 1.04·41-s + 0.487·43-s + 0.350·45-s − 0.479·47-s − 0.0521·49-s − 1.92·51-s + 0.441·53-s − 0.753·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.715266467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.715266467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 1.95T + 3T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 7 | \( 1 - 2.57T + 7T^{2} \) |
| 11 | \( 1 + 1.95T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 - 7.03T + 17T^{2} \) |
| 19 | \( 1 + 4.97T + 19T^{2} \) |
| 23 | \( 1 - 8.60T + 23T^{2} \) |
| 31 | \( 1 + 4.53T + 31T^{2} \) |
| 37 | \( 1 + 7.71T + 37T^{2} \) |
| 41 | \( 1 - 6.68T + 41T^{2} \) |
| 43 | \( 1 - 3.19T + 43T^{2} \) |
| 47 | \( 1 + 3.29T + 47T^{2} \) |
| 53 | \( 1 - 3.21T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 7.03T + 61T^{2} \) |
| 67 | \( 1 - 7.82T + 67T^{2} \) |
| 71 | \( 1 + 6.03T + 71T^{2} \) |
| 73 | \( 1 - 1.64T + 73T^{2} \) |
| 79 | \( 1 - 0.620T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 7.39T + 89T^{2} \) |
| 97 | \( 1 + 4.75T + 97T^{2} \) |
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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
−9.277450218516081309281801215743, −8.523754363514195477664504077305, −7.61212131801730967436553791999, −6.62540985501931558328196477084, −5.80541163010881588120352848137, −5.39231558464124073928098874732, −4.72529166086627709161211246551, −3.27300639358456998032096684771, −1.98397774110137084730563680820, −1.00433778352735841558578854291,
1.00433778352735841558578854291, 1.98397774110137084730563680820, 3.27300639358456998032096684771, 4.72529166086627709161211246551, 5.39231558464124073928098874732, 5.80541163010881588120352848137, 6.62540985501931558328196477084, 7.61212131801730967436553791999, 8.523754363514195477664504077305, 9.277450218516081309281801215743
