L(s) = 1 | + (0.707 + 0.707i)5-s − 2.89i·7-s + (1.84 + 1.84i)11-s + (−3.08 + 3.08i)13-s − 7.29·17-s + (1.23 − 1.23i)19-s − 4.60i·23-s + 1.00i·25-s + (−4.24 + 4.24i)29-s − 2.06·31-s + (2.04 − 2.04i)35-s + (−1.17 − 1.17i)37-s − 4.61i·41-s + (−3.03 − 3.03i)43-s − 11.7·47-s + ⋯ |
L(s) = 1 | + (0.316 + 0.316i)5-s − 1.09i·7-s + (0.556 + 0.556i)11-s + (−0.854 + 0.854i)13-s − 1.77·17-s + (0.283 − 0.283i)19-s − 0.960i·23-s + 0.200i·25-s + (−0.788 + 0.788i)29-s − 0.370·31-s + (0.345 − 0.345i)35-s + (−0.193 − 0.193i)37-s − 0.720i·41-s + (−0.462 − 0.462i)43-s − 1.70·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.06778253656\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06778253656\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + 2.89iT - 7T^{2} \) |
| 11 | \( 1 + (-1.84 - 1.84i)T + 11iT^{2} \) |
| 13 | \( 1 + (3.08 - 3.08i)T - 13iT^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 + (-1.23 + 1.23i)T - 19iT^{2} \) |
| 23 | \( 1 + 4.60iT - 23T^{2} \) |
| 29 | \( 1 + (4.24 - 4.24i)T - 29iT^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 + (1.17 + 1.17i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.61iT - 41T^{2} \) |
| 43 | \( 1 + (3.03 + 3.03i)T + 43iT^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + (2.73 + 2.73i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.11 - 3.11i)T + 59iT^{2} \) |
| 61 | \( 1 + (-2.34 + 2.34i)T - 61iT^{2} \) |
| 67 | \( 1 + (8.24 - 8.24i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.25iT - 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 0.113T + 79T^{2} \) |
| 83 | \( 1 + (-9.76 + 9.76i)T - 83iT^{2} \) |
| 89 | \( 1 - 3.74iT - 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−8.523517902905895516004239886239, −7.28255214401060292112720168994, −6.97642827816822024928401628873, −6.42492419179388552056618393142, −5.11854781446139218514358977266, −4.42315769781968269754695850500, −3.75474450361853223256478215660, −2.46504777677459082513935238839, −1.64351981339133080891802300466, −0.01962655068962892935756534003,
1.64459809199121765980465991820, 2.54067344937029512148273207329, 3.46931206701683487132360319203, 4.62232604458471513527765307823, 5.35035174842688412456805994209, 6.03000980043489404995151774444, 6.74097653424956504400361338847, 7.82025977755266884453985164498, 8.405266675986677629776010703947, 9.371279219151564112325387445558