L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 1.69·7-s − 8-s + 9-s + 10-s + 4.55·11-s − 12-s − 1.69·14-s + 15-s + 16-s + 2.35·17-s − 18-s + 6.51·19-s − 20-s − 1.69·21-s − 4.55·22-s + 8.94·23-s + 24-s + 25-s − 27-s + 1.69·28-s + 9.07·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.639·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.37·11-s − 0.288·12-s − 0.452·14-s + 0.258·15-s + 0.250·16-s + 0.571·17-s − 0.235·18-s + 1.49·19-s − 0.223·20-s − 0.369·21-s − 0.971·22-s + 1.86·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.319·28-s + 1.68·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.551471547\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.551471547\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 1.69T + 7T^{2} \) |
| 11 | \( 1 - 4.55T + 11T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 - 6.51T + 19T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 - 9.07T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 + 6.18T + 37T^{2} \) |
| 41 | \( 1 + 3.00T + 41T^{2} \) |
| 43 | \( 1 + 4.93T + 43T^{2} \) |
| 47 | \( 1 - 4.28T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 + 4.32T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 3.24T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 6.72T + 73T^{2} \) |
| 79 | \( 1 - 5.67T + 79T^{2} \) |
| 83 | \( 1 - 7.71T + 83T^{2} \) |
| 89 | \( 1 - 9.12T + 89T^{2} \) |
| 97 | \( 1 - 4.40T + 97T^{2} \) |
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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
−8.168806784012249371944182084368, −7.61851437321752240062015864551, −6.66575327570663301926197889298, −6.48358192850996901928129779511, −5.13650691792435437914904068013, −4.80660490673680163759695187625, −3.61970400401836408871970337608, −2.87386017577320539823888137076, −1.33674256028349186237311815988, −0.947557516638854379840405046474,
0.947557516638854379840405046474, 1.33674256028349186237311815988, 2.87386017577320539823888137076, 3.61970400401836408871970337608, 4.80660490673680163759695187625, 5.13650691792435437914904068013, 6.48358192850996901928129779511, 6.66575327570663301926197889298, 7.61851437321752240062015864551, 8.168806784012249371944182084368