| L(s) = 1 | + 2·5-s − 7·7-s − 52·11-s + 86·13-s + 30·17-s − 4·19-s − 120·23-s − 121·25-s − 246·29-s + 80·31-s − 14·35-s − 290·37-s + 374·41-s + 164·43-s − 464·47-s + 49·49-s + 162·53-s − 104·55-s − 180·59-s − 666·61-s + 172·65-s − 628·67-s − 296·71-s − 518·73-s + 364·77-s − 1.18e3·79-s − 220·83-s + ⋯ |
| L(s) = 1 | + 0.178·5-s − 0.377·7-s − 1.42·11-s + 1.83·13-s + 0.428·17-s − 0.0482·19-s − 1.08·23-s − 0.967·25-s − 1.57·29-s + 0.463·31-s − 0.0676·35-s − 1.28·37-s + 1.42·41-s + 0.581·43-s − 1.44·47-s + 1/7·49-s + 0.419·53-s − 0.254·55-s − 0.397·59-s − 1.39·61-s + 0.328·65-s − 1.14·67-s − 0.494·71-s − 0.830·73-s + 0.538·77-s − 1.68·79-s − 0.290·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 52 T + p^{3} T^{2} \) |
| 13 | \( 1 - 86 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 246 T + p^{3} T^{2} \) |
| 31 | \( 1 - 80 T + p^{3} T^{2} \) |
| 37 | \( 1 + 290 T + p^{3} T^{2} \) |
| 41 | \( 1 - 374 T + p^{3} T^{2} \) |
| 43 | \( 1 - 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 464 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 180 T + p^{3} T^{2} \) |
| 61 | \( 1 + 666 T + p^{3} T^{2} \) |
| 67 | \( 1 + 628 T + p^{3} T^{2} \) |
| 71 | \( 1 + 296 T + p^{3} T^{2} \) |
| 73 | \( 1 + 518 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1184 T + p^{3} T^{2} \) |
| 83 | \( 1 + 220 T + p^{3} T^{2} \) |
| 89 | \( 1 - 774 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1086 T + p^{3} T^{2} \) |
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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
−10.18164960530625225793776467768, −9.187703954651022086694602026648, −8.205686707207820759102499728021, −7.48246981861647231133555465905, −6.08786342656413109508038088136, −5.61619673568214684720296412422, −4.13093593505157228069684459461, −3.10377552181832227707634005925, −1.70183366795958560234134712703, 0,
1.70183366795958560234134712703, 3.10377552181832227707634005925, 4.13093593505157228069684459461, 5.61619673568214684720296412422, 6.08786342656413109508038088136, 7.48246981861647231133555465905, 8.205686707207820759102499728021, 9.187703954651022086694602026648, 10.18164960530625225793776467768
