L(s) = 1 | − 0.495·2-s + 1.43·3-s − 1.75·4-s + 2.92·5-s − 0.710·6-s − 7-s + 1.85·8-s − 0.942·9-s − 1.44·10-s − 11-s − 2.51·12-s − 4.48·13-s + 0.495·14-s + 4.19·15-s + 2.58·16-s + 4.95·17-s + 0.466·18-s − 2.23·19-s − 5.12·20-s − 1.43·21-s + 0.495·22-s − 5.09·23-s + 2.66·24-s + 3.53·25-s + 2.22·26-s − 5.65·27-s + 1.75·28-s + ⋯ |
L(s) = 1 | − 0.350·2-s + 0.828·3-s − 0.877·4-s + 1.30·5-s − 0.289·6-s − 0.377·7-s + 0.657·8-s − 0.314·9-s − 0.457·10-s − 0.301·11-s − 0.726·12-s − 1.24·13-s + 0.132·14-s + 1.08·15-s + 0.647·16-s + 1.20·17-s + 0.109·18-s − 0.513·19-s − 1.14·20-s − 0.313·21-s + 0.105·22-s − 1.06·23-s + 0.544·24-s + 0.706·25-s + 0.435·26-s − 1.08·27-s + 0.331·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 + 0.495T + 2T^{2} \) |
| 3 | \( 1 - 1.43T + 3T^{2} \) |
| 5 | \( 1 - 2.92T + 5T^{2} \) |
| 13 | \( 1 + 4.48T + 13T^{2} \) |
| 17 | \( 1 - 4.95T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 + 5.09T + 23T^{2} \) |
| 29 | \( 1 - 3.69T + 29T^{2} \) |
| 31 | \( 1 + 0.428T + 31T^{2} \) |
| 37 | \( 1 + 4.72T + 37T^{2} \) |
| 41 | \( 1 + 2.83T + 41T^{2} \) |
| 47 | \( 1 + 0.0467T + 47T^{2} \) |
| 53 | \( 1 - 1.04T + 53T^{2} \) |
| 59 | \( 1 + 4.32T + 59T^{2} \) |
| 61 | \( 1 + 8.64T + 61T^{2} \) |
| 67 | \( 1 - 7.58T + 67T^{2} \) |
| 71 | \( 1 - 2.25T + 71T^{2} \) |
| 73 | \( 1 + 5.58T + 73T^{2} \) |
| 79 | \( 1 + 2.02T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−8.307603920687139034187094407264, −7.84328652849151548586658759749, −6.87464091688348944961341354550, −5.78549308298119676624934836787, −5.36197178169403954293953447464, −4.37570617308589341359178019001, −3.31399337484451759452577603932, −2.51285034385190202803800335850, −1.60858746539119519476950210130, 0,
1.60858746539119519476950210130, 2.51285034385190202803800335850, 3.31399337484451759452577603932, 4.37570617308589341359178019001, 5.36197178169403954293953447464, 5.78549308298119676624934836787, 6.87464091688348944961341354550, 7.84328652849151548586658759749, 8.307603920687139034187094407264