L(s) = 1 | − 1.41i·3-s + (−1.22 + 1.87i)5-s − 2.64·7-s + 0.999·9-s + 3.46i·11-s − 2.44·13-s + (2.64 + 1.73i)15-s + 4.89·17-s − 6.48·19-s + 3.74i·21-s + (−2 − 4.58i)25-s − 5.65i·27-s + 6·29-s + 4.89·33-s + (3.24 − 4.94i)35-s + ⋯ |
L(s) = 1 | − 0.816i·3-s + (−0.547 + 0.836i)5-s − 0.999·7-s + 0.333·9-s + 1.04i·11-s − 0.679·13-s + (0.683 + 0.447i)15-s + 1.18·17-s − 1.48·19-s + 0.816i·21-s + (−0.400 − 0.916i)25-s − 1.08i·27-s + 1.11·29-s + 0.852·33-s + (0.547 − 0.836i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6706728837\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6706728837\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.22 - 1.87i)T \) |
| 7 | \( 1 + 2.64T \) |
good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 6.48T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.16iT - 37T^{2} \) |
| 41 | \( 1 + 7.48iT - 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 9.16iT - 53T^{2} \) |
| 59 | \( 1 + 6.48T + 59T^{2} \) |
| 61 | \( 1 + 11.2iT - 61T^{2} \) |
| 67 | \( 1 - 5.29T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 9.79T + 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 - 7.48iT - 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−8.652279580040091103030634396395, −7.65790893438675239638843332814, −7.23146299663790689498125059550, −6.64383172069713695415698174088, −5.93862861033862172121747233209, −4.60736524683186378276474366364, −3.79553325412865842628480730782, −2.75940640643767444313608650189, −1.92327341501673583207851110366, −0.25665601424364381782163696270,
1.13132792148628240227659842032, 2.89822475844004953790450008513, 3.64326979235844894914728658937, 4.45672768480111713636461524531, 5.15314827023502324563502369179, 6.12780902161366905196840274759, 6.91003677821265947653986522831, 8.016269366985007138305121725605, 8.541546041935512455551711585630, 9.375564852071783045931755790586