| L(s) = 1 | + 3-s − 0.876·5-s − 4.02·7-s + 9-s − 2.35·11-s − 3.89·13-s − 0.876·15-s − 3.87·17-s − 0.474·19-s − 4.02·21-s + 3.18·23-s − 4.23·25-s + 27-s + 4.66·29-s + 3.21·31-s − 2.35·33-s + 3.53·35-s − 0.756·37-s − 3.89·39-s + 5.43·41-s + 5.09·43-s − 0.876·45-s + 6.76·47-s + 9.23·49-s − 3.87·51-s + 13.4·53-s + 2.06·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.392·5-s − 1.52·7-s + 0.333·9-s − 0.709·11-s − 1.08·13-s − 0.226·15-s − 0.939·17-s − 0.108·19-s − 0.879·21-s + 0.663·23-s − 0.846·25-s + 0.192·27-s + 0.865·29-s + 0.577·31-s − 0.409·33-s + 0.597·35-s − 0.124·37-s − 0.624·39-s + 0.848·41-s + 0.776·43-s − 0.130·45-s + 0.987·47-s + 1.31·49-s − 0.542·51-s + 1.84·53-s + 0.278·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.166791102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.166791102\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 173 | \( 1 - T \) |
| good | 5 | \( 1 + 0.876T + 5T^{2} \) |
| 7 | \( 1 + 4.02T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 + 3.89T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 + 0.474T + 19T^{2} \) |
| 23 | \( 1 - 3.18T + 23T^{2} \) |
| 29 | \( 1 - 4.66T + 29T^{2} \) |
| 31 | \( 1 - 3.21T + 31T^{2} \) |
| 37 | \( 1 + 0.756T + 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 - 5.09T + 43T^{2} \) |
| 47 | \( 1 - 6.76T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + 6.84T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 4.71T + 67T^{2} \) |
| 71 | \( 1 - 1.21T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 0.856T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 5.76T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.485630458956428231556523925857, −7.57937645164023578685003784794, −7.07110570013600159032915923995, −6.36535006745613523601594216995, −5.46319261608875750351289282899, −4.47730890053595013918050645417, −3.78680070638205950331467811966, −2.76258271942830302289762999852, −2.40770195949574082122698725644, −0.55931193897518111479361117733,
0.55931193897518111479361117733, 2.40770195949574082122698725644, 2.76258271942830302289762999852, 3.78680070638205950331467811966, 4.47730890053595013918050645417, 5.46319261608875750351289282899, 6.36535006745613523601594216995, 7.07110570013600159032915923995, 7.57937645164023578685003784794, 8.485630458956428231556523925857
