| L(s) = 1 | + (1.75 − 1.57i)2-s + (−0.821 + 1.84i)3-s + (0.371 − 3.53i)4-s + (−0.00617 + 0.0290i)5-s + (1.47 + 4.52i)6-s + (−2.21 − 1.45i)7-s + (−2.15 − 2.96i)8-s + (−0.720 − 0.800i)9-s + (0.0350 + 0.0606i)10-s + (−2.88 + 1.64i)11-s + (6.21 + 3.58i)12-s + (−1.27 + 3.91i)13-s + (−6.16 + 0.945i)14-s + (−0.0485 − 0.0352i)15-s + (−1.49 − 0.318i)16-s + (2.61 − 2.90i)17-s + ⋯ |
| L(s) = 1 | + (1.23 − 1.11i)2-s + (−0.474 + 1.06i)3-s + (0.185 − 1.76i)4-s + (−0.00276 + 0.0130i)5-s + (0.600 + 1.84i)6-s + (−0.836 − 0.548i)7-s + (−0.761 − 1.04i)8-s + (−0.240 − 0.266i)9-s + (0.0110 + 0.0191i)10-s + (−0.869 + 0.494i)11-s + (1.79 + 1.03i)12-s + (−0.352 + 1.08i)13-s + (−1.64 + 0.252i)14-s + (−0.0125 − 0.00910i)15-s + (−0.374 − 0.0795i)16-s + (0.633 − 0.703i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27588 - 0.491655i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27588 - 0.491655i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.21 + 1.45i)T \) |
| 11 | \( 1 + (2.88 - 1.64i)T \) |
| good | 2 | \( 1 + (-1.75 + 1.57i)T + (0.209 - 1.98i)T^{2} \) |
| 3 | \( 1 + (0.821 - 1.84i)T + (-2.00 - 2.22i)T^{2} \) |
| 5 | \( 1 + (0.00617 - 0.0290i)T + (-4.56 - 2.03i)T^{2} \) |
| 13 | \( 1 + (1.27 - 3.91i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.61 + 2.90i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.627 + 5.97i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.70 + 2.96i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.60 + 3.58i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.713 - 3.35i)T + (-28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (9.81 - 4.37i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (-4.39 + 3.19i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 4.40iT - 43T^{2} \) |
| 47 | \( 1 + (1.98 - 0.208i)T + (45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (1.63 - 0.346i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-3.96 - 0.416i)T + (57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (8.45 + 1.79i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-1.46 - 2.53i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.813 + 2.50i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.203 + 1.93i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (8.27 - 7.45i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-2.78 - 8.56i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.57 - 0.909i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.7 + 3.49i)T + (78.4 + 57.0i)T^{2} \) |
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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
−14.09349229375332202226816812494, −13.18212516104524943633527496242, −12.19485497585540180759204611843, −11.05626895111823794927203914566, −10.32382502387076827858925180208, −9.452979757967799376604195199000, −6.89436040172117333069717550519, −5.11764340276677124150315756755, −4.41568861592333082636262297242, −2.92034968782207777782942243141,
3.29858984065232930128895741630, 5.46271077857523047172521014215, 6.07194282749874993210915316328, 7.25935760679059650384364844045, 8.231132619698732749069425004055, 10.32618979845549496710003040029, 12.25524728011144631431373317485, 12.62184185722948006173062299077, 13.37836798530933969307301059446, 14.59192896124951690215639806975
