| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s − 11-s + 12-s − 4·13-s − 14-s − 15-s + 16-s + 17-s + 18-s − 4·19-s − 20-s − 21-s − 22-s + 9·23-s + 24-s + 25-s − 4·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 1.10·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.218·21-s − 0.213·22-s + 1.87·23-s + 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s − 0.188·28-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
| \(L(1)\) |
\(\approx\) |
\(3.301465201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.301465201\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−8.030257398913232189803002704724, −7.33382313209062392605141927548, −6.80135953175214326491776380827, −6.02530208352056655717009468303, −4.92585896359353835890872169819, −4.60375570166123854504813266837, −3.60704281248655958524458674879, −2.86031748848033635070142768484, −2.29681095847263832742928275674, −0.837178258852899661424906608139,
0.837178258852899661424906608139, 2.29681095847263832742928275674, 2.86031748848033635070142768484, 3.60704281248655958524458674879, 4.60375570166123854504813266837, 4.92585896359353835890872169819, 6.02530208352056655717009468303, 6.80135953175214326491776380827, 7.33382313209062392605141927548, 8.030257398913232189803002704724
