L(s) = 1 | − 1.98i·2-s + 9.68i·3-s + 4.04·4-s + (10.8 + 2.74i)5-s + 19.2·6-s + 25.5i·7-s − 23.9i·8-s − 66.8·9-s + (5.45 − 21.5i)10-s − 14.6·11-s + 39.1i·12-s − 70.2i·13-s + 50.8·14-s + (−26.5 + 104. i)15-s − 15.2·16-s + 105. i·17-s + ⋯ |
L(s) = 1 | − 0.703i·2-s + 1.86i·3-s + 0.505·4-s + (0.969 + 0.245i)5-s + 1.31·6-s + 1.38i·7-s − 1.05i·8-s − 2.47·9-s + (0.172 − 0.681i)10-s − 0.400·11-s + 0.942i·12-s − 1.49i·13-s + 0.970·14-s + (−0.457 + 1.80i)15-s − 0.238·16-s + 1.50i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.58522 + 1.23385i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58522 + 1.23385i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-10.8 - 2.74i)T \) |
| 23 | \( 1 + 23iT \) |
good | 2 | \( 1 + 1.98iT - 8T^{2} \) |
| 3 | \( 1 - 9.68iT - 27T^{2} \) |
| 7 | \( 1 - 25.5iT - 343T^{2} \) |
| 11 | \( 1 + 14.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 70.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 105. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 75.6T + 6.85e3T^{2} \) |
| 29 | \( 1 + 144.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 53.1iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 23.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 142. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 190. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 58.7iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 342.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 459.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 282. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 308.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 520. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 713.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.19e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 35.1T + 7.04e5T^{2} \) |
| 97 | \( 1 + 561. iT - 9.12e5T^{2} \) |
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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
−13.12834356306475289328768915034, −11.99451792530826144099402656865, −10.85027872735652173203573675809, −10.25622672091893410040983988614, −9.538192263594409907932048911230, −8.388804979925214759995931046491, −5.96935844702018918636235823779, −5.33565707803368387200772155806, −3.42463094557071879461173317674, −2.48163262957400714662974225872,
1.15828389482635961553218462367, 2.45028435304449667741385091188, 5.31345064083616356667447537094, 6.61194331669900481503984692695, 7.07978597333924177359466067489, 7.951606242819408712156727794959, 9.433452531860973435080858251251, 11.12078400385872173829986132296, 11.91517306546320342444510619625, 13.23696200681225808997394703694