L(s) = 1 | + 0.583·3-s − 2.69·5-s + 4.05·7-s − 2.66·9-s + 1.48·11-s + 1.08·13-s − 1.57·15-s − 6.87·17-s − 0.887·19-s + 2.36·21-s − 6.60·23-s + 2.27·25-s − 3.29·27-s + 8.49·29-s + 1.83·31-s + 0.864·33-s − 10.9·35-s + 8.02·37-s + 0.631·39-s − 4.78·41-s + 4.02·43-s + 7.17·45-s + 11.7·47-s + 9.46·49-s − 4.00·51-s − 8.04·53-s − 3.99·55-s + ⋯ |
L(s) = 1 | + 0.336·3-s − 1.20·5-s + 1.53·7-s − 0.886·9-s + 0.446·11-s + 0.300·13-s − 0.405·15-s − 1.66·17-s − 0.203·19-s + 0.516·21-s − 1.37·23-s + 0.454·25-s − 0.635·27-s + 1.57·29-s + 0.330·31-s + 0.150·33-s − 1.84·35-s + 1.31·37-s + 0.101·39-s − 0.747·41-s + 0.613·43-s + 1.06·45-s + 1.71·47-s + 1.35·49-s − 0.561·51-s − 1.10·53-s − 0.538·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.706244525\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.706244525\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 - 0.583T + 3T^{2} \) |
| 5 | \( 1 + 2.69T + 5T^{2} \) |
| 7 | \( 1 - 4.05T + 7T^{2} \) |
| 11 | \( 1 - 1.48T + 11T^{2} \) |
| 13 | \( 1 - 1.08T + 13T^{2} \) |
| 17 | \( 1 + 6.87T + 17T^{2} \) |
| 19 | \( 1 + 0.887T + 19T^{2} \) |
| 23 | \( 1 + 6.60T + 23T^{2} \) |
| 29 | \( 1 - 8.49T + 29T^{2} \) |
| 31 | \( 1 - 1.83T + 31T^{2} \) |
| 37 | \( 1 - 8.02T + 37T^{2} \) |
| 41 | \( 1 + 4.78T + 41T^{2} \) |
| 43 | \( 1 - 4.02T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 8.04T + 53T^{2} \) |
| 59 | \( 1 - 4.98T + 59T^{2} \) |
| 61 | \( 1 - 5.28T + 61T^{2} \) |
| 67 | \( 1 + 1.14T + 67T^{2} \) |
| 71 | \( 1 - 9.07T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.131402345461844294143162325357, −7.68845382771992413968232304936, −6.70765620137839220948931961913, −6.01422725673056785248843352866, −4.99473798575308669590150987601, −4.27530649581224240274997980535, −3.92208004156632223190663106544, −2.69570319819037293715181169353, −1.96922630611875749599055253792, −0.66678069352980227188403088671,
0.66678069352980227188403088671, 1.96922630611875749599055253792, 2.69570319819037293715181169353, 3.92208004156632223190663106544, 4.27530649581224240274997980535, 4.99473798575308669590150987601, 6.01422725673056785248843352866, 6.70765620137839220948931961913, 7.68845382771992413968232304936, 8.131402345461844294143162325357