| L(s) = 1 | − 2.68·2-s − 3.08·3-s + 5.18·4-s + 8.26·6-s + 7-s − 8.52·8-s + 6.51·9-s − 1.03·11-s − 15.9·12-s + 4.03·13-s − 2.68·14-s + 12.4·16-s − 4.11·17-s − 17.4·18-s − 6.23·19-s − 3.08·21-s + 2.77·22-s + 23-s + 26.3·24-s − 10.8·26-s − 10.8·27-s + 5.18·28-s − 8.14·29-s + 10.0·31-s − 16.4·32-s + 3.19·33-s + 11.0·34-s + ⋯ |
| L(s) = 1 | − 1.89·2-s − 1.78·3-s + 2.59·4-s + 3.37·6-s + 0.377·7-s − 3.01·8-s + 2.17·9-s − 0.312·11-s − 4.61·12-s + 1.11·13-s − 0.716·14-s + 3.12·16-s − 0.998·17-s − 4.11·18-s − 1.43·19-s − 0.672·21-s + 0.592·22-s + 0.208·23-s + 5.36·24-s − 2.11·26-s − 2.08·27-s + 0.979·28-s − 1.51·29-s + 1.80·31-s − 2.90·32-s + 0.556·33-s + 1.89·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2741230261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2741230261\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 3 | \( 1 + 3.08T + 3T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 - 4.03T + 13T^{2} \) |
| 17 | \( 1 + 4.11T + 17T^{2} \) |
| 19 | \( 1 + 6.23T + 19T^{2} \) |
| 29 | \( 1 + 8.14T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 4.74T + 37T^{2} \) |
| 41 | \( 1 + 4.45T + 41T^{2} \) |
| 43 | \( 1 - 4.71T + 43T^{2} \) |
| 47 | \( 1 + 4.57T + 47T^{2} \) |
| 53 | \( 1 - 6.63T + 53T^{2} \) |
| 59 | \( 1 - 5.25T + 59T^{2} \) |
| 61 | \( 1 + 0.922T + 61T^{2} \) |
| 67 | \( 1 - 9.91T + 67T^{2} \) |
| 71 | \( 1 + 3.55T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 4.18T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 + 0.694T + 97T^{2} \) |
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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
−8.543781558152191159439654073071, −7.78511780287843223583101077624, −6.90486638416169736714821866392, −6.44886488801745683196867687426, −5.90418562147942395327240313584, −4.95214145493252605370308832849, −3.89987706393368789738009802419, −2.29636723415882888996273573758, −1.44320873923169767705746728778, −0.46254075301310614596539713011,
0.46254075301310614596539713011, 1.44320873923169767705746728778, 2.29636723415882888996273573758, 3.89987706393368789738009802419, 4.95214145493252605370308832849, 5.90418562147942395327240313584, 6.44886488801745683196867687426, 6.90486638416169736714821866392, 7.78511780287843223583101077624, 8.543781558152191159439654073071
