L(s) = 1 | + 0.334·3-s + 1.98i·5-s − i·7-s − 2.88·9-s + i·11-s + (−2.44 − 2.65i)13-s + 0.664i·15-s − 5.84·17-s + 2.86i·19-s − 0.334i·21-s + 0.564·23-s + 1.04·25-s − 1.96·27-s + 8.26·29-s − 1.77i·31-s + ⋯ |
L(s) = 1 | + 0.192·3-s + 0.889i·5-s − 0.377i·7-s − 0.962·9-s + 0.301i·11-s + (−0.677 − 0.735i)13-s + 0.171i·15-s − 1.41·17-s + 0.658i·19-s − 0.0728i·21-s + 0.117·23-s + 0.208·25-s − 0.378·27-s + 1.53·29-s − 0.317i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.258480446\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.258480446\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (2.44 + 2.65i)T \) |
good | 3 | \( 1 - 0.334T + 3T^{2} \) |
| 5 | \( 1 - 1.98iT - 5T^{2} \) |
| 17 | \( 1 + 5.84T + 17T^{2} \) |
| 19 | \( 1 - 2.86iT - 19T^{2} \) |
| 23 | \( 1 - 0.564T + 23T^{2} \) |
| 29 | \( 1 - 8.26T + 29T^{2} \) |
| 31 | \( 1 + 1.77iT - 31T^{2} \) |
| 37 | \( 1 - 0.265iT - 37T^{2} \) |
| 41 | \( 1 + 5.63iT - 41T^{2} \) |
| 43 | \( 1 + 0.932T + 43T^{2} \) |
| 47 | \( 1 + 10.4iT - 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 8.63iT - 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 1.98iT - 67T^{2} \) |
| 71 | \( 1 + 10.5iT - 71T^{2} \) |
| 73 | \( 1 + 3.74iT - 73T^{2} \) |
| 79 | \( 1 - 4.24T + 79T^{2} \) |
| 83 | \( 1 + 8.15iT - 83T^{2} \) |
| 89 | \( 1 + 18.0iT - 89T^{2} \) |
| 97 | \( 1 + 12.1iT - 97T^{2} \) |
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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
−8.465979132742207240604973580636, −7.49899560745778720760916694669, −6.95938065725382470026267844968, −6.24594276008378044014029258862, −5.40695808180982662479538770748, −4.54641649803523170224788040852, −3.59313032483552382695288950686, −2.77063560131021997204006486134, −2.14735599429111218578474462550, −0.42179918456329904758092609867,
0.903828986076521221328624855859, 2.28752108907896395610581335456, 2.84497253617170422166334295201, 4.12981466477653580758556856580, 4.83739672324836004814496736411, 5.39779663074453622410005247775, 6.46019317579518847489332743987, 6.90908207377607833969319008421, 8.197572849358985555778143010711, 8.492635255366390395037343553644