| L(s) = 1 | + 1.35i·2-s + 0.163·4-s + (−1.94 + 1.10i)5-s − 1.26i·7-s + 2.93i·8-s + (−1.49 − 2.63i)10-s + 0.474·11-s + 0.407i·13-s + 1.72·14-s − 3.64·16-s + 2.97i·17-s − 5.80·19-s + (−0.318 + 0.180i)20-s + 0.643i·22-s + 3.06i·23-s + ⋯ |
| L(s) = 1 | + 0.958i·2-s + 0.0818·4-s + (−0.870 + 0.492i)5-s − 0.479i·7-s + 1.03i·8-s + (−0.472 − 0.833i)10-s + 0.143·11-s + 0.113i·13-s + 0.459·14-s − 0.911·16-s + 0.720i·17-s − 1.33·19-s + (−0.0711 + 0.0403i)20-s + 0.137i·22-s + 0.639i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7757683582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7757683582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.94 - 1.10i)T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.35iT - 2T^{2} \) |
| 7 | \( 1 + 1.26iT - 7T^{2} \) |
| 11 | \( 1 - 0.474T + 11T^{2} \) |
| 13 | \( 1 - 0.407iT - 13T^{2} \) |
| 17 | \( 1 - 2.97iT - 17T^{2} \) |
| 19 | \( 1 + 5.80T + 19T^{2} \) |
| 23 | \( 1 - 3.06iT - 23T^{2} \) |
| 31 | \( 1 + 4.49T + 31T^{2} \) |
| 37 | \( 1 - 7.20iT - 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 - 1.03iT - 43T^{2} \) |
| 47 | \( 1 - 2.97iT - 47T^{2} \) |
| 53 | \( 1 + 7.87iT - 53T^{2} \) |
| 59 | \( 1 + 4.45T + 59T^{2} \) |
| 61 | \( 1 + 1.71T + 61T^{2} \) |
| 67 | \( 1 + 15.7iT - 67T^{2} \) |
| 71 | \( 1 + 9.00T + 71T^{2} \) |
| 73 | \( 1 - 0.677iT - 73T^{2} \) |
| 79 | \( 1 + 1.63T + 79T^{2} \) |
| 83 | \( 1 - 15.5iT - 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 7.10iT - 97T^{2} \) |
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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
−10.28908752254057469160006012372, −8.984979715253227900340335292967, −8.178604265867083484677843429161, −7.66845794165569010368190133552, −6.74149422833650455489349073239, −6.38000835302904600822366426698, −5.18685053707555532412780307017, −4.19144777720079824863696876313, −3.26927950508633624254477542566, −1.88452240575834175204841029215,
0.30119890612260922712815559865, 1.77611952244692809886065153407, 2.82944837680644355067929172673, 3.82301553239242156167596907932, 4.59569177059350889882412422680, 5.74557124499697868970183343380, 6.83791274989606842037590866465, 7.53424354421027107836004707461, 8.683621413572599606433107203709, 9.080988416906721371545225378461
