| L(s) = 1 | − 1.48i·2-s − 3-s − 0.193·4-s − 4.15i·5-s + 1.48i·6-s − 2.67i·8-s + 9-s − 6.15·10-s + 3.19i·11-s + 0.193·12-s + (−1.48 + 3.28i)13-s + 4.15i·15-s − 4.35·16-s − 3.35·17-s − 1.48i·18-s + 2.38i·19-s + ⋯ |
| L(s) = 1 | − 1.04i·2-s − 0.577·3-s − 0.0969·4-s − 1.85i·5-s + 0.604i·6-s − 0.945i·8-s + 0.333·9-s − 1.94·10-s + 0.963i·11-s + 0.0559·12-s + (−0.410 + 0.911i)13-s + 1.07i·15-s − 1.08·16-s − 0.812·17-s − 0.349i·18-s + 0.547i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1383194752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1383194752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (1.48 - 3.28i)T \) |
| good | 2 | \( 1 + 1.48iT - 2T^{2} \) |
| 5 | \( 1 + 4.15iT - 5T^{2} \) |
| 11 | \( 1 - 3.19iT - 11T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 - 2.38iT - 19T^{2} \) |
| 23 | \( 1 - 0.387T + 23T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 - 10.7iT - 31T^{2} \) |
| 37 | \( 1 - 1.61iT - 37T^{2} \) |
| 41 | \( 1 - 1.45iT - 41T^{2} \) |
| 43 | \( 1 - 1.92T + 43T^{2} \) |
| 47 | \( 1 + 3.76iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 6.15iT - 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + 5.61iT - 67T^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 15.6iT - 73T^{2} \) |
| 79 | \( 1 + 8.96T + 79T^{2} \) |
| 83 | \( 1 - 6.99iT - 83T^{2} \) |
| 89 | \( 1 - 0.932iT - 89T^{2} \) |
| 97 | \( 1 + 3.35iT - 97T^{2} \) |
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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
−9.087624368439693920090233367051, −7.83479066128507489051001126712, −6.98219337473295574473882349987, −6.07359402085860734872736703763, −4.84735414181332234720780930689, −4.61065575765795872003783602654, −3.59481036421459030761546156561, −1.91661352727798043243506738908, −1.52017363996875493607568444759, −0.05006247155408423926986687019,
2.29290838490209376026516058917, 3.03213426754001186191992816323, 4.21289177338175511976256952723, 5.59903555944447002956213963203, 5.94880250836217116333678227625, 6.66678857614951176514905661678, 7.42373397788222276127540892779, 7.76390828717565526411808748743, 8.924906288237989465269232271030, 9.941918735166031664149252935758
