| L(s) = 1 | − 0.435·2-s − 1.81·4-s − 4.22·5-s − 2.90·7-s + 1.65·8-s + 1.83·10-s − 0.559·11-s + 1.50·13-s + 1.26·14-s + 2.89·16-s + 6.57·19-s + 7.64·20-s + 0.243·22-s + 1.98·23-s + 12.8·25-s − 0.653·26-s + 5.26·28-s + 2.24·29-s − 2.41·31-s − 4.58·32-s + 12.2·35-s + 4.90·37-s − 2.86·38-s − 7.00·40-s − 7.01·41-s + 5.38·43-s + 1.01·44-s + ⋯ |
| L(s) = 1 | − 0.307·2-s − 0.905·4-s − 1.88·5-s − 1.09·7-s + 0.586·8-s + 0.581·10-s − 0.168·11-s + 0.416·13-s + 0.338·14-s + 0.724·16-s + 1.50·19-s + 1.70·20-s + 0.0519·22-s + 0.414·23-s + 2.56·25-s − 0.128·26-s + 0.995·28-s + 0.417·29-s − 0.433·31-s − 0.809·32-s + 2.07·35-s + 0.806·37-s − 0.464·38-s − 1.10·40-s − 1.09·41-s + 0.821·43-s + 0.152·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.435T + 2T^{2} \) |
| 5 | \( 1 + 4.22T + 5T^{2} \) |
| 7 | \( 1 + 2.90T + 7T^{2} \) |
| 11 | \( 1 + 0.559T + 11T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 19 | \( 1 - 6.57T + 19T^{2} \) |
| 23 | \( 1 - 1.98T + 23T^{2} \) |
| 29 | \( 1 - 2.24T + 29T^{2} \) |
| 31 | \( 1 + 2.41T + 31T^{2} \) |
| 37 | \( 1 - 4.90T + 37T^{2} \) |
| 41 | \( 1 + 7.01T + 41T^{2} \) |
| 43 | \( 1 - 5.38T + 43T^{2} \) |
| 47 | \( 1 + 7.70T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 0.354T + 59T^{2} \) |
| 61 | \( 1 + 4.96T + 61T^{2} \) |
| 67 | \( 1 + 3.56T + 67T^{2} \) |
| 71 | \( 1 - 3.72T + 71T^{2} \) |
| 73 | \( 1 - 8.61T + 73T^{2} \) |
| 79 | \( 1 + 2.73T + 79T^{2} \) |
| 83 | \( 1 + 9.16T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.448730916331775177567727993575, −7.79595042369438087785519135964, −7.26251452650811526624060603133, −6.32159035962046725658110199914, −5.13381985168945127356757719382, −4.45136006218563457605141965814, −3.46002484639015166020376970217, −3.21513898578496158303837290689, −0.985254837824371413532214063994, 0,
0.985254837824371413532214063994, 3.21513898578496158303837290689, 3.46002484639015166020376970217, 4.45136006218563457605141965814, 5.13381985168945127356757719382, 6.32159035962046725658110199914, 7.26251452650811526624060603133, 7.79595042369438087785519135964, 8.448730916331775177567727993575
