L(s) = 1 | + (2.57 − 2.57i)5-s − 3.74·7-s + (3.64 + 3.64i)11-s + (4.64 − 4.64i)13-s + 0.913i·17-s + (−1.41 − 1.41i)19-s − 6i·23-s − 8.29i·25-s + (−1.66 − 1.66i)29-s + 6.57i·31-s + (−9.64 + 9.64i)35-s + (1 + i)37-s + 0.913·41-s + (3.74 − 3.74i)43-s + 6·47-s + ⋯ |
L(s) = 1 | + (1.15 − 1.15i)5-s − 1.41·7-s + (1.09 + 1.09i)11-s + (1.28 − 1.28i)13-s + 0.221i·17-s + (−0.324 − 0.324i)19-s − 1.25i·23-s − 1.65i·25-s + (−0.309 − 0.309i)29-s + 1.18i·31-s + (−1.63 + 1.63i)35-s + (0.164 + 0.164i)37-s + 0.142·41-s + (0.570 − 0.570i)43-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.009738977\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.009738977\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.57 + 2.57i)T - 5iT^{2} \) |
| 7 | \( 1 + 3.74T + 7T^{2} \) |
| 11 | \( 1 + (-3.64 - 3.64i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.64 + 4.64i)T - 13iT^{2} \) |
| 17 | \( 1 - 0.913iT - 17T^{2} \) |
| 19 | \( 1 + (1.41 + 1.41i)T + 19iT^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + (1.66 + 1.66i)T + 29iT^{2} \) |
| 31 | \( 1 - 6.57iT - 31T^{2} \) |
| 37 | \( 1 + (-1 - i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.913T + 41T^{2} \) |
| 43 | \( 1 + (-3.74 + 3.74i)T - 43iT^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + (3.49 - 3.49i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.29 + 1.29i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.291 - 0.291i)T - 61iT^{2} \) |
| 67 | \( 1 + (3.32 + 3.32i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.2iT - 71T^{2} \) |
| 73 | \( 1 - 8.58iT - 73T^{2} \) |
| 79 | \( 1 + 8.39iT - 79T^{2} \) |
| 83 | \( 1 + (-4.93 + 4.93i)T - 83iT^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−9.001175014457068808624639437655, −8.374734342134625459403473522239, −7.11848377766867819681576213873, −6.22622192552727381508844141159, −5.96563829618015298444915699442, −4.87568722552054005896917073822, −4.01802349751789394270214718961, −2.98221451158870843895957503911, −1.78526910200935647211683132615, −0.73973057298285860734421868995,
1.33537633040776321577615069321, 2.48656670274356158176206960745, 3.49257545136141387275396622572, 3.91366501433822325026812077715, 5.73765315502582087447116682272, 6.18385424368909313910343721231, 6.54449051608197696198354565728, 7.35102491330650031304820514924, 8.699691552095985477963023801210, 9.399127975652889506924442706111