| L(s) = 1 | − 1.35·2-s − 3-s − 0.175·4-s + 1.35·6-s + 1.59·7-s + 2.93·8-s + 9-s + 3.33·11-s + 0.175·12-s + 7.05·13-s − 2.15·14-s − 3.61·16-s + 4.09·17-s − 1.35·18-s + 0.567·19-s − 1.59·21-s − 4.50·22-s + 6.30·23-s − 2.93·24-s − 9.52·26-s − 27-s − 0.279·28-s − 2.78·29-s − 0.995·31-s − 0.988·32-s − 3.33·33-s − 5.53·34-s + ⋯ |
| L(s) = 1 | − 0.955·2-s − 0.577·3-s − 0.0876·4-s + 0.551·6-s + 0.603·7-s + 1.03·8-s + 0.333·9-s + 1.00·11-s + 0.0505·12-s + 1.95·13-s − 0.576·14-s − 0.904·16-s + 0.993·17-s − 0.318·18-s + 0.130·19-s − 0.348·21-s − 0.959·22-s + 1.31·23-s − 0.599·24-s − 1.86·26-s − 0.192·27-s − 0.0528·28-s − 0.516·29-s − 0.178·31-s − 0.174·32-s − 0.580·33-s − 0.948·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.083115349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.083115349\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 7 | \( 1 - 1.59T + 7T^{2} \) |
| 11 | \( 1 - 3.33T + 11T^{2} \) |
| 13 | \( 1 - 7.05T + 13T^{2} \) |
| 17 | \( 1 - 4.09T + 17T^{2} \) |
| 19 | \( 1 - 0.567T + 19T^{2} \) |
| 23 | \( 1 - 6.30T + 23T^{2} \) |
| 29 | \( 1 + 2.78T + 29T^{2} \) |
| 31 | \( 1 + 0.995T + 31T^{2} \) |
| 37 | \( 1 + 3.55T + 37T^{2} \) |
| 41 | \( 1 - 1.16T + 41T^{2} \) |
| 43 | \( 1 - 0.117T + 43T^{2} \) |
| 47 | \( 1 - 7.64T + 47T^{2} \) |
| 53 | \( 1 - 0.523T + 53T^{2} \) |
| 59 | \( 1 - 0.983T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 5.55T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 5.02T + 83T^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 + 1.70T + 97T^{2} \) |
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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
−8.974202738068428167742934981653, −8.738805915098542627708178618202, −7.76025972263796058152352709409, −7.03923523733820074434378734471, −6.08358302792928319859791001929, −5.27690454012064343135306359959, −4.27464149691434021308747497577, −3.47464187506563361008315325574, −1.53941946295797567691624508938, −0.987353738408260084385826513754,
0.987353738408260084385826513754, 1.53941946295797567691624508938, 3.47464187506563361008315325574, 4.27464149691434021308747497577, 5.27690454012064343135306359959, 6.08358302792928319859791001929, 7.03923523733820074434378734471, 7.76025972263796058152352709409, 8.738805915098542627708178618202, 8.974202738068428167742934981653
