| L(s) = 1 | − 3·3-s − 6·5-s + 9·9-s − 12·11-s − 38·13-s + 18·15-s + 126·17-s + 20·19-s − 168·23-s − 89·25-s − 27·27-s + 30·29-s − 88·31-s + 36·33-s + 254·37-s + 114·39-s − 42·41-s + 52·43-s − 54·45-s − 96·47-s − 378·51-s + 198·53-s + 72·55-s − 60·57-s − 660·59-s + 538·61-s + 228·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.536·5-s + 1/3·9-s − 0.328·11-s − 0.810·13-s + 0.309·15-s + 1.79·17-s + 0.241·19-s − 1.52·23-s − 0.711·25-s − 0.192·27-s + 0.192·29-s − 0.509·31-s + 0.189·33-s + 1.12·37-s + 0.468·39-s − 0.159·41-s + 0.184·43-s − 0.178·45-s − 0.297·47-s − 1.03·51-s + 0.513·53-s + 0.176·55-s − 0.139·57-s − 1.45·59-s + 1.12·61-s + 0.435·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9907839337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9907839337\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 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
−8.462879875562792656225754457777, −7.62235251118918036705968347911, −7.40067362365243112279254453722, −6.08708566127395806914103144772, −5.60735786508346364122185059879, −4.66252435836434547247680646503, −3.85279386700909339738299278950, −2.88875801231923841669416617752, −1.66328457108609785484450342861, −0.45788628322029784585915869234,
0.45788628322029784585915869234, 1.66328457108609785484450342861, 2.88875801231923841669416617752, 3.85279386700909339738299278950, 4.66252435836434547247680646503, 5.60735786508346364122185059879, 6.08708566127395806914103144772, 7.40067362365243112279254453722, 7.62235251118918036705968347911, 8.462879875562792656225754457777
