| L(s) = 1 | − i·3-s + 2.44i·7-s − 9-s − 0.576·11-s − 6.44i·13-s + 1.89i·17-s − 8.27·19-s + 2.44·21-s − 4.20i·23-s + i·27-s − 6.53·29-s + 4.86·31-s + 0.576i·33-s − 0.218i·37-s − 6.44·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.923i·7-s − 0.333·9-s − 0.173·11-s − 1.78i·13-s + 0.460i·17-s − 1.89·19-s + 0.533·21-s − 0.877i·23-s + 0.192i·27-s − 1.21·29-s + 0.874·31-s + 0.100i·33-s − 0.0358i·37-s − 1.03·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7258959698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7258959698\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.44iT - 7T^{2} \) |
| 11 | \( 1 + 0.576T + 11T^{2} \) |
| 13 | \( 1 + 6.44iT - 13T^{2} \) |
| 17 | \( 1 - 1.89iT - 17T^{2} \) |
| 19 | \( 1 + 8.27T + 19T^{2} \) |
| 23 | \( 1 + 4.20iT - 23T^{2} \) |
| 29 | \( 1 + 6.53T + 29T^{2} \) |
| 31 | \( 1 - 4.86T + 31T^{2} \) |
| 37 | \( 1 + 0.218iT - 37T^{2} \) |
| 41 | \( 1 - 6.45T + 41T^{2} \) |
| 43 | \( 1 - 3.42iT - 43T^{2} \) |
| 47 | \( 1 + 9.61iT - 47T^{2} \) |
| 53 | \( 1 - 13.9iT - 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 4.81T + 61T^{2} \) |
| 67 | \( 1 - 3.87iT - 67T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 - 10.0iT - 73T^{2} \) |
| 79 | \( 1 - 4.74T + 79T^{2} \) |
| 83 | \( 1 + 3.43iT - 83T^{2} \) |
| 89 | \( 1 - 5.68T + 89T^{2} \) |
| 97 | \( 1 - 6.58iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.140552153744976260756558618417, −7.47604682118029732693167505542, −6.56502122846931478390022608822, −5.91968067877170690069686736816, −5.52970163266991581990436208321, −4.56297173582212840879590134865, −3.66967801038344339516270735626, −2.58328568553047254090896385515, −2.29277315577140163256398786651, −0.949590402788653191836919974214,
0.19173676756976984743450533930, 1.62313512913916772985424520967, 2.42720325791886668942985597710, 3.60180997725571734422319271988, 4.20987680706043477586918707632, 4.60415305033541529715242552404, 5.58975944295651436608889846369, 6.45117878421904871141663831344, 6.94126748780689865022134644009, 7.69491114565718109754498322159
