L(s) = 1 | + i·3-s + (−0.311 + 2.21i)5-s − i·7-s − 9-s − 3.80·11-s + 0.622i·13-s + (−2.21 − 0.311i)15-s + 4.42i·17-s − 0.622·19-s + 21-s + 2.62i·23-s + (−4.80 − 1.37i)25-s − i·27-s − 9.61·29-s − 0.622·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.139 + 0.990i)5-s − 0.377i·7-s − 0.333·9-s − 1.14·11-s + 0.172i·13-s + (−0.571 − 0.0803i)15-s + 1.07i·17-s − 0.142·19-s + 0.218·21-s + 0.546i·23-s + (−0.961 − 0.275i)25-s − 0.192i·27-s − 1.78·29-s − 0.111·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0490200 + 0.701230i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0490200 + 0.701230i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.311 - 2.21i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 - 0.622iT - 13T^{2} \) |
| 17 | \( 1 - 4.42iT - 17T^{2} \) |
| 19 | \( 1 + 0.622T + 19T^{2} \) |
| 23 | \( 1 - 2.62iT - 23T^{2} \) |
| 29 | \( 1 + 9.61T + 29T^{2} \) |
| 31 | \( 1 + 0.622T + 31T^{2} \) |
| 37 | \( 1 - 1.24iT - 37T^{2} \) |
| 41 | \( 1 - 4.62T + 41T^{2} \) |
| 43 | \( 1 + 4.85iT - 43T^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 - 13.4iT - 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 8.10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 2.56T + 71T^{2} \) |
| 73 | \( 1 - 10.9iT - 73T^{2} \) |
| 79 | \( 1 - 6.75T + 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 - 8.23T + 89T^{2} \) |
| 97 | \( 1 + 4.23iT - 97T^{2} \) |
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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
−10.69029386765897057589502338697, −9.964694587237382782858953682193, −9.026654417780999242493102146341, −7.896980252371559972761966036414, −7.33897802293293312042760266843, −6.18858648818175526707249723758, −5.37686732087160642271244737514, −4.11625025672061864171498245697, −3.34407461035736248915611616000, −2.13860795247112511023962789916,
0.32127839289216802624767038425, 1.92036459569631477302163167282, 3.08629570439020468749418510957, 4.59060295905927345824442514648, 5.34818871555658923351092578168, 6.19130714841340144958861850226, 7.55374534757610520793900952326, 7.911082904077350346884410076530, 8.991622236445638461718038906066, 9.544396058514262573705764290329