L(s) = 1 | − 3·3-s + 1.44·5-s + 9·9-s + 46.1·11-s + 32.2·13-s − 4.33·15-s − 77.7·17-s − 12.6·19-s − 100.·23-s − 122.·25-s − 27·27-s + 213.·29-s − 42.0·31-s − 138.·33-s + 310.·37-s − 96.6·39-s − 44.0·41-s + 381.·43-s + 13.0·45-s + 358.·47-s + 233.·51-s + 184.·53-s + 66.7·55-s + 37.9·57-s − 454.·59-s − 11.8·61-s + 46.6·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.129·5-s + 0.333·9-s + 1.26·11-s + 0.687·13-s − 0.0746·15-s − 1.10·17-s − 0.152·19-s − 0.914·23-s − 0.983·25-s − 0.192·27-s + 1.36·29-s − 0.243·31-s − 0.729·33-s + 1.37·37-s − 0.397·39-s − 0.167·41-s + 1.35·43-s + 0.0431·45-s + 1.11·47-s + 0.640·51-s + 0.479·53-s + 0.163·55-s + 0.0882·57-s − 1.00·59-s − 0.0248·61-s + 0.0889·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.718179194\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.718179194\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.44T + 125T^{2} \) |
| 11 | \( 1 - 46.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 77.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 12.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 213.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 42.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 310.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 44.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 381.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 358.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 184.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 454.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 11.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 590.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 494.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 975.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 299.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.40e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 695.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 481.T + 9.12e5T^{2} \) |
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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
−10.37255822845713354277932402440, −9.409120945323936111059425179743, −8.661748678793344177324007853422, −7.53927439954336379683445581646, −6.38499611495421019577083349170, −6.02025377994200025568597487104, −4.55970182936209625355235175717, −3.82145389378921496583651543780, −2.14867448465467280991806039652, −0.828516135274339562149944945450,
0.828516135274339562149944945450, 2.14867448465467280991806039652, 3.82145389378921496583651543780, 4.55970182936209625355235175717, 6.02025377994200025568597487104, 6.38499611495421019577083349170, 7.53927439954336379683445581646, 8.661748678793344177324007853422, 9.409120945323936111059425179743, 10.37255822845713354277932402440