| L(s) = 1 | − 3.69i·5-s − 3.88i·7-s + 4.98i·11-s + 0.561·13-s + (3.56 − 2.07i)17-s + 4.03·19-s − 1.09i·23-s − 8.68·25-s − 1.62i·29-s − 6.07i·31-s − 14.3·35-s − 5.77i·37-s − 9.47i·41-s − 10.3·43-s + 8.07·47-s + ⋯ |
| L(s) = 1 | − 1.65i·5-s − 1.46i·7-s + 1.50i·11-s + 0.155·13-s + (0.863 − 0.503i)17-s + 0.926·19-s − 0.227i·23-s − 1.73·25-s − 0.301i·29-s − 1.09i·31-s − 2.43·35-s − 0.949i·37-s − 1.47i·41-s − 1.57·43-s + 1.17·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 + 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.863 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.775481389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.775481389\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 + (-3.56 + 2.07i)T \) |
| good | 5 | \( 1 + 3.69iT - 5T^{2} \) |
| 7 | \( 1 + 3.88iT - 7T^{2} \) |
| 11 | \( 1 - 4.98iT - 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 19 | \( 1 - 4.03T + 19T^{2} \) |
| 23 | \( 1 + 1.09iT - 23T^{2} \) |
| 29 | \( 1 + 1.62iT - 29T^{2} \) |
| 31 | \( 1 + 6.07iT - 31T^{2} \) |
| 37 | \( 1 + 5.77iT - 37T^{2} \) |
| 41 | \( 1 + 9.47iT - 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 8.07T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 - 6.30T + 59T^{2} \) |
| 61 | \( 1 + 9.93iT - 61T^{2} \) |
| 67 | \( 1 - 6.30T + 67T^{2} \) |
| 71 | \( 1 + 1.70iT - 71T^{2} \) |
| 73 | \( 1 - 10.6iT - 73T^{2} \) |
| 79 | \( 1 - 6.07iT - 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 0.910iT - 97T^{2} \) |
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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
−7.80804698204940471782472258611, −7.42491227693912760175846762451, −6.71696979674913740307081930266, −5.37883931997004185595302868400, −5.12857179623728727378674500376, −4.07455382498024604083327265484, −3.88617151823536535087654257015, −2.25899477426226467036584664533, −1.23156911302275164100924568510, −0.53023079757336957584212986095,
1.38853359897784033960765358786, 2.68361522364961547807232091260, 3.08050353468470913314977832597, 3.69007982755554557185389516369, 5.24110727492152842813194677179, 5.71850228105286639006897727174, 6.36421697130793036293074252181, 6.97624172919802115120699086594, 7.961785771272034157637246663083, 8.446393381058713608871362743070
