L(s) = 1 | − 3·3-s + 5-s + 6·9-s − 11-s + 3·13-s − 3·15-s + 17-s − 19-s − 4·23-s + 25-s − 9·27-s + 7·29-s − 31-s + 3·33-s + 8·37-s − 9·39-s + 12·41-s − 10·43-s + 6·45-s + 13·47-s − 7·49-s − 3·51-s − 53-s − 55-s + 3·57-s − 3·59-s − 11·61-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 0.447·5-s + 2·9-s − 0.301·11-s + 0.832·13-s − 0.774·15-s + 0.242·17-s − 0.229·19-s − 0.834·23-s + 1/5·25-s − 1.73·27-s + 1.29·29-s − 0.179·31-s + 0.522·33-s + 1.31·37-s − 1.44·39-s + 1.87·41-s − 1.52·43-s + 0.894·45-s + 1.89·47-s − 49-s − 0.420·51-s − 0.137·53-s − 0.134·55-s + 0.397·57-s − 0.390·59-s − 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.394271864\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.394271864\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−14.28514765808193, −13.61194476358725, −13.27217834941689, −12.67097223224077, −12.15935768777674, −11.88696745804266, −11.15967820251891, −10.76793025006916, −10.47120969109957, −9.833435281539582, −9.380730767137837, −8.657415826826015, −7.938846280461548, −7.492662978349149, −6.709178548979945, −6.183132355210596, −5.988574558017713, −5.431049461131036, −4.640851003801094, −4.419918001049208, −3.558787932136284, −2.710633526793128, −1.867565680799298, −1.112517613817206, −0.5243410820684251,
0.5243410820684251, 1.112517613817206, 1.867565680799298, 2.710633526793128, 3.558787932136284, 4.419918001049208, 4.640851003801094, 5.431049461131036, 5.988574558017713, 6.183132355210596, 6.709178548979945, 7.492662978349149, 7.938846280461548, 8.657415826826015, 9.380730767137837, 9.833435281539582, 10.47120969109957, 10.76793025006916, 11.15967820251891, 11.88696745804266, 12.15935768777674, 12.67097223224077, 13.27217834941689, 13.61194476358725, 14.28514765808193