| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 2.37·7-s − 8-s + 9-s − 10-s + 11-s − 12-s − 4·13-s − 2.37·14-s − 15-s + 16-s + 17-s − 18-s + 4·19-s + 20-s − 2.37·21-s − 22-s − 1.62·23-s + 24-s + 25-s + 4·26-s − 27-s + 2.37·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.896·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 1.10·13-s − 0.634·14-s − 0.258·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.517·21-s − 0.213·22-s − 0.339·23-s + 0.204·24-s + 0.200·25-s + 0.784·26-s − 0.192·27-s + 0.448·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.319021669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.319021669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2.37T + 7T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 1.62T + 23T^{2} \) |
| 29 | \( 1 + 6.37T + 29T^{2} \) |
| 31 | \( 1 + 9.11T + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 4.74T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 2.74T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−8.058304830513310511418852477823, −7.35483143834538702932315332468, −7.02548105609422570947197930081, −5.79943890746369625948709625261, −5.51266963798826842399347047975, −4.63164546312937028143441318362, −3.69746589021308049170562338408, −2.45410394658772014578618962496, −1.73917000213990662085948766465, −0.71719443631182514220201277305,
0.71719443631182514220201277305, 1.73917000213990662085948766465, 2.45410394658772014578618962496, 3.69746589021308049170562338408, 4.63164546312937028143441318362, 5.51266963798826842399347047975, 5.79943890746369625948709625261, 7.02548105609422570947197930081, 7.35483143834538702932315332468, 8.058304830513310511418852477823
