L(s) = 1 | + (−1.63 + 1.63i)3-s + (−0.707 − 0.707i)5-s − i·7-s − 2.32i·9-s + (0.230 + 0.230i)11-s + (−1.57 + 1.57i)13-s + 2.30·15-s + 4.13·17-s + (−0.553 + 0.553i)19-s + (1.63 + 1.63i)21-s + 0.651i·23-s + 1.00i·25-s + (−1.09 − 1.09i)27-s + (−5.41 + 5.41i)29-s − 4.74·31-s + ⋯ |
L(s) = 1 | + (−0.942 + 0.942i)3-s + (−0.316 − 0.316i)5-s − 0.377i·7-s − 0.776i·9-s + (0.0694 + 0.0694i)11-s + (−0.437 + 0.437i)13-s + 0.595·15-s + 1.00·17-s + (−0.127 + 0.127i)19-s + (0.356 + 0.356i)21-s + 0.135i·23-s + 0.200i·25-s + (−0.211 − 0.211i)27-s + (−1.00 + 1.00i)29-s − 0.851·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0169 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0169 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3674548778\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3674548778\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (1.63 - 1.63i)T - 3iT^{2} \) |
| 11 | \( 1 + (-0.230 - 0.230i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.57 - 1.57i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.13T + 17T^{2} \) |
| 19 | \( 1 + (0.553 - 0.553i)T - 19iT^{2} \) |
| 23 | \( 1 - 0.651iT - 23T^{2} \) |
| 29 | \( 1 + (5.41 - 5.41i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.74T + 31T^{2} \) |
| 37 | \( 1 + (-4.34 - 4.34i)T + 37iT^{2} \) |
| 41 | \( 1 + 6.29iT - 41T^{2} \) |
| 43 | \( 1 + (1.77 + 1.77i)T + 43iT^{2} \) |
| 47 | \( 1 - 2.06T + 47T^{2} \) |
| 53 | \( 1 + (3.94 + 3.94i)T + 53iT^{2} \) |
| 59 | \( 1 + (10.2 + 10.2i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.32 - 3.32i)T - 61iT^{2} \) |
| 67 | \( 1 + (-10.7 + 10.7i)T - 67iT^{2} \) |
| 71 | \( 1 - 2.53iT - 71T^{2} \) |
| 73 | \( 1 + 10.8iT - 73T^{2} \) |
| 79 | \( 1 + 6.25T + 79T^{2} \) |
| 83 | \( 1 + (4.72 - 4.72i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.25iT - 89T^{2} \) |
| 97 | \( 1 + 2.47T + 97T^{2} \) |
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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
−9.117088750092971246287437472320, −7.985368164299426544220889782339, −7.32287021330684338910505656124, −6.36075926746489942612638544172, −5.43247165964044757379381124233, −4.92923685294439739998256167619, −4.06743428987168816345261991540, −3.34639171506722317193738473638, −1.69385274718689124714443971387, −0.16565133958671237831873712479,
1.09016063082199111789902526058, 2.31804121821414194552385862978, 3.39188991478519306724337305024, 4.53478556397849056273287121081, 5.70743495029049353154736887691, 5.90249092855766320554741175204, 6.98211962687963568519472245346, 7.53631240434769475584214329224, 8.173627775600571785087654140330, 9.300622499946275765776227046916