| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 4·11-s − 12-s − 15-s + 16-s + 2·17-s − 18-s − 8·19-s + 20-s − 4·22-s − 23-s + 24-s + 25-s − 27-s − 6·29-s + 30-s − 8·31-s − 32-s − 4·33-s − 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.83·19-s + 0.223·20-s − 0.852·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.182·30-s − 1.43·31-s − 0.176·32-s − 0.696·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6947235138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6947235138\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.50439859512022, −12.95778901793907, −12.52327281381063, −12.20824985292713, −11.38161927601580, −11.19401757274424, −10.74264155914922, −10.15390099527500, −9.540890198250934, −9.385592037942738, −8.749873967442303, −8.182740097847389, −7.719213395766926, −7.017686892038115, −6.479703541013055, −6.282449795157575, −5.691680242078442, −4.980582163209193, −4.482587200550167, −3.633715073093688, −3.384489850827215, −2.205173901357506, −1.818571365974972, −1.296702396672494, −0.2986635609857109,
0.2986635609857109, 1.296702396672494, 1.818571365974972, 2.205173901357506, 3.384489850827215, 3.633715073093688, 4.482587200550167, 4.980582163209193, 5.691680242078442, 6.282449795157575, 6.479703541013055, 7.017686892038115, 7.719213395766926, 8.182740097847389, 8.749873967442303, 9.385592037942738, 9.540890198250934, 10.15390099527500, 10.74264155914922, 11.19401757274424, 11.38161927601580, 12.20824985292713, 12.52327281381063, 12.95778901793907, 13.50439859512022
