| L(s) = 1 | − 1.17i·3-s − 1.90i·7-s + 1.61·9-s + 3.80·11-s − 1.23i·13-s − 2i·17-s + 6.15·19-s − 2.23·21-s − 1.90i·23-s − 5.42i·27-s − 3.61·29-s − 0.898·31-s − 4.47i·33-s − 9.70i·37-s − 1.45·39-s + ⋯ |
| L(s) = 1 | − 0.678i·3-s − 0.718i·7-s + 0.539·9-s + 1.14·11-s − 0.342i·13-s − 0.485i·17-s + 1.41·19-s − 0.487·21-s − 0.396i·23-s − 1.04i·27-s − 0.671·29-s − 0.161·31-s − 0.778i·33-s − 1.59i·37-s − 0.232·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.298643876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.298643876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.17iT - 3T^{2} \) |
| 7 | \( 1 + 1.90iT - 7T^{2} \) |
| 11 | \( 1 - 3.80T + 11T^{2} \) |
| 13 | \( 1 + 1.23iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 6.15T + 19T^{2} \) |
| 23 | \( 1 + 1.90iT - 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 0.898T + 31T^{2} \) |
| 37 | \( 1 + 9.70iT - 37T^{2} \) |
| 41 | \( 1 - 0.854T + 41T^{2} \) |
| 43 | \( 1 - 11.1iT - 43T^{2} \) |
| 47 | \( 1 - 9.68iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 - 3.80T + 59T^{2} \) |
| 61 | \( 1 - 1.38T + 61T^{2} \) |
| 67 | \( 1 - 1.45iT - 67T^{2} \) |
| 71 | \( 1 - 9.06T + 71T^{2} \) |
| 73 | \( 1 - 6.76iT - 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 4.25iT - 83T^{2} \) |
| 89 | \( 1 - 3.09T + 89T^{2} \) |
| 97 | \( 1 + 14.1iT - 97T^{2} \) |
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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
−7.953462630937507952811184385287, −7.50517379016886775508704672642, −6.92082622386750915773764926383, −6.23193320396536503944740675729, −5.34344319264267434810573642829, −4.33539925988078962487446680140, −3.72758492168214222430167203896, −2.65915326158009069955581425347, −1.45298808475920547287780713387, −0.78590897549567538238694366799,
1.21915368975139066175758718452, 2.17421026116636202656925130299, 3.55433104084113324025209976486, 3.83197935541718894506519066525, 4.98722792959635078350446134896, 5.46214376746942617035574346371, 6.49341484909456474117234029969, 7.06598791335246259794714975198, 7.964515557787612046999658105965, 8.881942718008516126645703889744
