| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 6·5-s − 6·6-s + 16·7-s + 8·8-s + 9·9-s + 12·10-s − 12·12-s − 38·13-s + 32·14-s − 18·15-s + 16·16-s + 126·17-s + 18·18-s − 20·19-s + 24·20-s − 48·21-s + 168·23-s − 24·24-s − 89·25-s − 76·26-s − 27·27-s + 64·28-s − 30·29-s − 36·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.536·5-s − 0.408·6-s + 0.863·7-s + 0.353·8-s + 1/3·9-s + 0.379·10-s − 0.288·12-s − 0.810·13-s + 0.610·14-s − 0.309·15-s + 1/4·16-s + 1.79·17-s + 0.235·18-s − 0.241·19-s + 0.268·20-s − 0.498·21-s + 1.52·23-s − 0.204·24-s − 0.711·25-s − 0.573·26-s − 0.192·27-s + 0.431·28-s − 0.192·29-s − 0.219·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.435183757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.435183757\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 126 T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 - 168 T + p^{3} T^{2} \) |
| 29 | \( 1 + 30 T + p^{3} T^{2} \) |
| 31 | \( 1 + 88 T + p^{3} T^{2} \) |
| 37 | \( 1 - 254 T + p^{3} T^{2} \) |
| 41 | \( 1 + 42 T + p^{3} T^{2} \) |
| 43 | \( 1 - 52 T + p^{3} T^{2} \) |
| 47 | \( 1 + 96 T + p^{3} T^{2} \) |
| 53 | \( 1 - 198 T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 - 538 T + p^{3} T^{2} \) |
| 67 | \( 1 - 884 T + p^{3} T^{2} \) |
| 71 | \( 1 - 792 T + p^{3} T^{2} \) |
| 73 | \( 1 + 218 T + p^{3} T^{2} \) |
| 79 | \( 1 - 520 T + p^{3} T^{2} \) |
| 83 | \( 1 - 492 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1154 T + p^{3} T^{2} \) |
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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
−10.12883466998197608511383819328, −9.379359469853129912224015041878, −7.986956521569031417211847675212, −7.32221240583361734653021339233, −6.23164333623958306999662893941, −5.32774016985699104218699981141, −4.85633366271146046784448283325, −3.54850621145690462271714946194, −2.22918951058934538772009941198, −1.05003389726335309247895753665,
1.05003389726335309247895753665, 2.22918951058934538772009941198, 3.54850621145690462271714946194, 4.85633366271146046784448283325, 5.32774016985699104218699981141, 6.23164333623958306999662893941, 7.32221240583361734653021339233, 7.986956521569031417211847675212, 9.379359469853129912224015041878, 10.12883466998197608511383819328
