| L(s) = 1 | + 1.71·3-s + 0.391·5-s + 7-s − 0.0460·9-s + 3.06·11-s − 2.78·13-s + 0.672·15-s − 6.22·17-s − 19-s + 1.71·21-s + 4.78·23-s − 4.84·25-s − 5.23·27-s − 7.82·29-s − 1.34·31-s + 5.26·33-s + 0.391·35-s + 3.82·37-s − 4.78·39-s − 4.39·41-s + 5.82·43-s − 0.0180·45-s + 5.71·47-s + 49-s − 10.6·51-s − 7.15·53-s + 1.19·55-s + ⋯ |
| L(s) = 1 | + 0.992·3-s + 0.175·5-s + 0.377·7-s − 0.0153·9-s + 0.923·11-s − 0.771·13-s + 0.173·15-s − 1.50·17-s − 0.229·19-s + 0.375·21-s + 0.997·23-s − 0.969·25-s − 1.00·27-s − 1.45·29-s − 0.241·31-s + 0.916·33-s + 0.0661·35-s + 0.629·37-s − 0.765·39-s − 0.685·41-s + 0.888·43-s − 0.00268·45-s + 0.834·47-s + 0.142·49-s − 1.49·51-s − 0.982·53-s + 0.161·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.71T + 3T^{2} \) |
| 5 | \( 1 - 0.391T + 5T^{2} \) |
| 11 | \( 1 - 3.06T + 11T^{2} \) |
| 13 | \( 1 + 2.78T + 13T^{2} \) |
| 17 | \( 1 + 6.22T + 17T^{2} \) |
| 23 | \( 1 - 4.78T + 23T^{2} \) |
| 29 | \( 1 + 7.82T + 29T^{2} \) |
| 31 | \( 1 + 1.34T + 31T^{2} \) |
| 37 | \( 1 - 3.82T + 37T^{2} \) |
| 41 | \( 1 + 4.39T + 41T^{2} \) |
| 43 | \( 1 - 5.82T + 43T^{2} \) |
| 47 | \( 1 - 5.71T + 47T^{2} \) |
| 53 | \( 1 + 7.15T + 53T^{2} \) |
| 59 | \( 1 + 7.17T + 59T^{2} \) |
| 61 | \( 1 + 8.50T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 0.935T + 71T^{2} \) |
| 73 | \( 1 + 4.12T + 73T^{2} \) |
| 79 | \( 1 + 2.95T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 1.06T + 89T^{2} \) |
| 97 | \( 1 - 4.50T + 97T^{2} \) |
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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
−7.50245093263258689196475894612, −6.91134600153018472074665403871, −6.10288505403354802736893225828, −5.35371292017920075780715973889, −4.41228469564613108722655616541, −3.92025522831354794269075256813, −2.95908749824610114709844453550, −2.24055631838275322536401146455, −1.57283399695533435773204712778, 0,
1.57283399695533435773204712778, 2.24055631838275322536401146455, 2.95908749824610114709844453550, 3.92025522831354794269075256813, 4.41228469564613108722655616541, 5.35371292017920075780715973889, 6.10288505403354802736893225828, 6.91134600153018472074665403871, 7.50245093263258689196475894612
