L(s) = 1 | + 2.23i·5-s + 4.47i·17-s + 4·19-s + 8.94i·23-s − 5.00·25-s − 8·31-s + 8.94i·47-s + 7·49-s + 4.47i·53-s + 2·61-s + 16·79-s − 17.8i·83-s − 10.0·85-s + 8.94i·95-s − 17.8i·107-s + ⋯ |
L(s) = 1 | + 0.999i·5-s + 1.08i·17-s + 0.917·19-s + 1.86i·23-s − 1.00·25-s − 1.43·31-s + 1.30i·47-s + 49-s + 0.614i·53-s + 0.256·61-s + 1.80·79-s − 1.96i·83-s − 1.08·85-s + 0.917i·95-s − 1.72i·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.938328 + 0.938328i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.938328 + 0.938328i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 8.94iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 8.94iT - 47T^{2} \) |
| 53 | \( 1 - 4.47iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 17.8iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−10.69240128540947547888921042899, −9.809889049743542898033321700471, −9.052279995459641796793851349931, −7.76945700073456537490153964855, −7.28839388055053732670904875648, −6.16109111081262327179434600241, −5.42347997177186152207552171764, −3.93389652753075287805071163372, −3.13257995320767604323059817500, −1.73149626659091747084411437283,
0.70582787609937990066954065305, 2.29395385364598507054374219129, 3.72273210085640936662163350749, 4.84193456827757688845139374979, 5.49565879879442986829035508867, 6.72879555656138451432287786711, 7.65105595114249053970818004956, 8.601726817612098631243085833497, 9.251801175202539443198554796721, 10.07802825330082104845248678599