| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 9-s − 11-s − 13-s − 14-s + 16-s − 17-s + 18-s − 22-s + 25-s − 26-s − 28-s + 32-s − 34-s + 36-s − 37-s − 41-s − 43-s − 44-s + 50-s − 52-s − 56-s + 59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 9-s − 11-s − 13-s − 14-s + 16-s − 17-s + 18-s − 22-s + 25-s − 26-s − 28-s + 32-s − 34-s + 36-s − 37-s − 41-s − 43-s − 44-s + 50-s − 52-s − 56-s + 59-s + 2·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.412935305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.412935305\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−11.36017534861238890667907724888, −10.29636103035951404129179399223, −9.860523315203318361915213464857, −8.394551369894446720600487578423, −7.05508314377897558225914319674, −6.79519899146886347540741399333, −5.37912258917513824504005023745, −4.56684429375797396990716617328, −3.34847841920329681632604645067, −2.23050369734682394136455039807,
2.23050369734682394136455039807, 3.34847841920329681632604645067, 4.56684429375797396990716617328, 5.37912258917513824504005023745, 6.79519899146886347540741399333, 7.05508314377897558225914319674, 8.394551369894446720600487578423, 9.860523315203318361915213464857, 10.29636103035951404129179399223, 11.36017534861238890667907724888
