L(s) = 1 | − 5-s − 3.13i·7-s − 2.04·11-s + 5.04·13-s + 2.17·17-s − 2.11i·19-s + (−4.01 − 2.62i)23-s + 25-s + 8.97i·29-s + 3.32·31-s + 3.13i·35-s + 2.50i·37-s + 8.54i·41-s + 10.3i·43-s + 11.6i·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.18i·7-s − 0.618·11-s + 1.39·13-s + 0.526·17-s − 0.485i·19-s + (−0.836 − 0.548i)23-s + 0.200·25-s + 1.66i·29-s + 0.597·31-s + 0.529i·35-s + 0.412i·37-s + 1.33i·41-s + 1.57i·43-s + 1.69i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.764093196\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.764093196\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + (4.01 + 2.62i)T \) |
good | 7 | \( 1 + 3.13iT - 7T^{2} \) |
| 11 | \( 1 + 2.04T + 11T^{2} \) |
| 13 | \( 1 - 5.04T + 13T^{2} \) |
| 17 | \( 1 - 2.17T + 17T^{2} \) |
| 19 | \( 1 + 2.11iT - 19T^{2} \) |
| 29 | \( 1 - 8.97iT - 29T^{2} \) |
| 31 | \( 1 - 3.32T + 31T^{2} \) |
| 37 | \( 1 - 2.50iT - 37T^{2} \) |
| 41 | \( 1 - 8.54iT - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 11.6iT - 47T^{2} \) |
| 53 | \( 1 - 2.92T + 53T^{2} \) |
| 59 | \( 1 - 0.993iT - 59T^{2} \) |
| 61 | \( 1 - 8.28iT - 61T^{2} \) |
| 67 | \( 1 + 11.8iT - 67T^{2} \) |
| 71 | \( 1 + 13.2iT - 71T^{2} \) |
| 73 | \( 1 - 9.03T + 73T^{2} \) |
| 79 | \( 1 + 5.58iT - 79T^{2} \) |
| 83 | \( 1 + 7.01T + 83T^{2} \) |
| 89 | \( 1 - 7.84T + 89T^{2} \) |
| 97 | \( 1 - 12.4iT - 97T^{2} \) |
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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
−7.964002375610561356909874999097, −7.15907769604313412658628353079, −6.46256165751861558216170378024, −5.87873841982149926084771843946, −4.75690768555951728152607420348, −4.37841964301721628478850223688, −3.42500972773010560258883586582, −2.95350556285035729435004687511, −1.50964829566534507806063994740, −0.77695782052713540013368993445,
0.58318523999300411432005790094, 1.91099324810051277857200576573, 2.56059715114246017222771818354, 3.68830953995695383809296858600, 3.98209415533555420073411364926, 5.32916762855262783408630389207, 5.61406311776620712452621174266, 6.28505806908705257005880070557, 7.17544129545986919063346795369, 7.982666636052642795224391852608