| L(s) = 1 | − 5i·5-s − 20.7i·7-s + 36.3·11-s − 47·13-s + 21i·17-s + 62.3i·19-s + 36.3·23-s − 25·25-s − 123i·29-s − 25.9i·31-s − 103.·35-s − 178·37-s − 342i·41-s − 233. i·43-s + 306.·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s − 1.12i·7-s + 0.996·11-s − 1.00·13-s + 0.299i·17-s + 0.752i·19-s + 0.329·23-s − 0.200·25-s − 0.787i·29-s − 0.150i·31-s − 0.501·35-s − 0.790·37-s − 1.30i·41-s − 0.829i·43-s + 0.951·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8901127041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8901127041\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5iT \) |
| good | 7 | \( 1 + 20.7iT - 343T^{2} \) |
| 11 | \( 1 - 36.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 47T + 2.19e3T^{2} \) |
| 17 | \( 1 - 21iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 62.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 36.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 123iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 25.9iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 178T + 5.06e4T^{2} \) |
| 41 | \( 1 + 342iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 233. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 306.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 414iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 446.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 542T + 2.26e5T^{2} \) |
| 67 | \( 1 + 155. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 852.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 232T + 3.89e5T^{2} \) |
| 79 | \( 1 + 348. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 405.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.35e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−8.349763197766835549413694720404, −7.43107958000274690565680051857, −6.95800716291271090520981687587, −5.97381611044847364163438242546, −5.05968999715628395222260700937, −4.13223003103368139136883782474, −3.63908584709686883285952056317, −2.19690992331681416789362765604, −1.17473480166577947928305689713, −0.18512213008751159904039247649,
1.33735204124623489846335512415, 2.50049631462422821294937239586, 3.10462251469429307071810747416, 4.35056165086576368470179048200, 5.15295650009407186980454866880, 5.99791061500127688941836083757, 6.82074837197660996824644662399, 7.40629457137832818037287898564, 8.496245740881756200056824224534, 9.148701862920846544285939254152
