L(s) = 1 | + i·3-s + i·5-s − 4·7-s − 9-s + 4i·11-s + 6i·13-s − 15-s + 2·17-s − 4i·19-s − 4i·21-s − 25-s − i·27-s + 10i·29-s + 4·31-s − 4·33-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.447i·5-s − 1.51·7-s − 0.333·9-s + 1.20i·11-s + 1.66i·13-s − 0.258·15-s + 0.485·17-s − 0.917i·19-s − 0.872i·21-s − 0.200·25-s − 0.192i·27-s + 1.85i·29-s + 0.718·31-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6162880382\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6162880382\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
good | 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.272678541084238934061553269664, −8.363211219324362430360272856636, −7.06819867990263348994392820584, −6.86659041855308834230744097731, −6.19902425730433669150703468634, −5.00923077211499261253285991722, −4.42569690049208014771951709504, −3.44670996225844450834996575427, −2.85248453130779436263171756528, −1.69878518591344983456696306147,
0.20768565453367446775216289522, 0.988903210979481225061074026822, 2.54998270868670808763388484055, 3.25337190636729185627102772255, 3.92151443716021661783707678190, 5.33768161526801806051953379402, 5.97660136606033441119944711772, 6.26850834084122763802501859717, 7.42314574897950223358127994066, 8.067379407392791323753526054529