| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 12-s + 13-s − 15-s + 16-s − 2·17-s − 18-s − 20-s − 8·23-s − 24-s + 25-s − 26-s + 27-s + 2·29-s + 30-s − 32-s + 2·34-s + 36-s + 6·37-s + 39-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 + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.223·20-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.371·29-s + 0.182·30-s − 0.176·32-s + 0.342·34-s + 1/6·36-s + 0.986·37-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5696724936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5696724936\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.43099133750153, −11.91708530263041, −11.52088752147315, −11.20598562893759, −10.52190679208870, −10.21566387011999, −9.668842343115725, −9.390155179858622, −8.773202012906928, −8.266693051197154, −8.080865643981522, −7.672068722689994, −7.003094776631237, −6.602935590323325, −6.174170312334706, −5.602179772495594, −4.914630234444600, −4.322302534968916, −3.976997987807063, −3.273030657621109, −2.896386604386259, −2.216301841547135, −1.690792362535025, −1.165401825541248, −0.2120069689738731,
0.2120069689738731, 1.165401825541248, 1.690792362535025, 2.216301841547135, 2.896386604386259, 3.273030657621109, 3.976997987807063, 4.322302534968916, 4.914630234444600, 5.602179772495594, 6.174170312334706, 6.602935590323325, 7.003094776631237, 7.672068722689994, 8.080865643981522, 8.266693051197154, 8.773202012906928, 9.390155179858622, 9.668842343115725, 10.21566387011999, 10.52190679208870, 11.20598562893759, 11.52088752147315, 11.91708530263041, 12.43099133750153
