| L(s) = 1 | + 2.30·2-s − 2.69·4-s − 33.7·7-s − 24.6·8-s − 11·11-s − 37.9·13-s − 77.6·14-s − 35.1·16-s − 44.2·17-s − 42.3·19-s − 25.3·22-s + 30.6·23-s − 87.2·26-s + 90.9·28-s − 194.·29-s − 82.9·31-s + 116.·32-s − 101.·34-s − 445.·37-s − 97.6·38-s − 206.·41-s − 197.·43-s + 29.6·44-s + 70.6·46-s − 275.·47-s + 793.·49-s + 102.·52-s + ⋯ |
| L(s) = 1 | + 0.814·2-s − 0.337·4-s − 1.82·7-s − 1.08·8-s − 0.301·11-s − 0.808·13-s − 1.48·14-s − 0.548·16-s − 0.631·17-s − 0.511·19-s − 0.245·22-s + 0.278·23-s − 0.658·26-s + 0.614·28-s − 1.24·29-s − 0.480·31-s + 0.641·32-s − 0.514·34-s − 1.97·37-s − 0.416·38-s − 0.785·41-s − 0.699·43-s + 0.101·44-s + 0.226·46-s − 0.854·47-s + 2.31·49-s + 0.272·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.03193029996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03193029996\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 2.30T + 8T^{2} \) |
| 7 | \( 1 + 33.7T + 343T^{2} \) |
| 13 | \( 1 + 37.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 42.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 30.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 82.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 445.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 206.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 197.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 275.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 382.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 771.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 210.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 452.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 310.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 174.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 183.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 613.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.20e3T + 9.12e5T^{2} \) |
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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
−8.885937498238222483369426169527, −7.67453053051997066565081989548, −6.72890706224350732785535133211, −6.27890551521085674511296698640, −5.34147195823161874972845061171, −4.66949474284767484596765377594, −3.55553217515586416399560302980, −3.21113167474916166330394123238, −2.08467609096589145973482208323, −0.06411821187917064835977661685,
0.06411821187917064835977661685, 2.08467609096589145973482208323, 3.21113167474916166330394123238, 3.55553217515586416399560302980, 4.66949474284767484596765377594, 5.34147195823161874972845061171, 6.27890551521085674511296698640, 6.72890706224350732785535133211, 7.67453053051997066565081989548, 8.885937498238222483369426169527
