L(s) = 1 | + (−0.651 + 1.25i)2-s − i·3-s + (−1.15 − 1.63i)4-s − i·5-s + (1.25 + 0.651i)6-s + 2.14·7-s + (2.80 − 0.381i)8-s − 9-s + (1.25 + 0.651i)10-s − 3.55·11-s + (−1.63 + 1.15i)12-s − 4.04·13-s + (−1.39 + 2.68i)14-s − 15-s + (−1.34 + 3.76i)16-s − 0.843i·17-s + ⋯ |
L(s) = 1 | + (−0.460 + 0.887i)2-s − 0.577i·3-s + (−0.575 − 0.817i)4-s − 0.447i·5-s + (0.512 + 0.265i)6-s + 0.808·7-s + (0.990 − 0.134i)8-s − 0.333·9-s + (0.396 + 0.205i)10-s − 1.07·11-s + (−0.471 + 0.332i)12-s − 1.12·13-s + (−0.372 + 0.718i)14-s − 0.258·15-s + (−0.336 + 0.941i)16-s − 0.204i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.03112211711\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03112211711\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.651 - 1.25i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (-2.71 + 3.95i)T \) |
good | 7 | \( 1 - 2.14T + 7T^{2} \) |
| 11 | \( 1 + 3.55T + 11T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 + 0.843iT - 17T^{2} \) |
| 19 | \( 1 + 5.13T + 19T^{2} \) |
| 29 | \( 1 - 0.275T + 29T^{2} \) |
| 31 | \( 1 - 8.41iT - 31T^{2} \) |
| 37 | \( 1 - 2.63iT - 37T^{2} \) |
| 41 | \( 1 - 7.36T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 10.4iT - 47T^{2} \) |
| 53 | \( 1 - 5.60iT - 53T^{2} \) |
| 59 | \( 1 - 2.95iT - 59T^{2} \) |
| 61 | \( 1 + 7.05iT - 61T^{2} \) |
| 67 | \( 1 + 2.62T + 67T^{2} \) |
| 71 | \( 1 - 2.73iT - 71T^{2} \) |
| 73 | \( 1 - 4.84T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 + 9.45T + 83T^{2} \) |
| 89 | \( 1 + 16.0iT - 89T^{2} \) |
| 97 | \( 1 - 0.845iT - 97T^{2} \) |
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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
−9.880168066799239806340062734678, −8.801452470858973559585641069748, −8.315280734755494670857866087678, −7.60511238865977457719956763113, −6.91773494425639809109779137528, −5.96632792113198312738699629625, −4.92760540729040092915548640048, −4.64433241361889635022813422785, −2.65597704318236732541107360095, −1.46897223764933387402725365305,
0.01443731131908163227947297039, 1.99057299230121877459740052090, 2.74886878954138924999036347116, 3.90217083963044764871793665258, 4.76493016360955726205094047236, 5.51163268901462849062607665116, 7.01201360422629652185233572797, 7.86298309280635626120091970391, 8.397742621649563498816669660214, 9.428913570257308268064441740109