L(s) = 1 | + 0.437·2-s − 3-s − 1.80·4-s − 0.437·6-s − 7-s − 1.66·8-s + 9-s − 11-s + 1.80·12-s + 0.0257·13-s − 0.437·14-s + 2.88·16-s + 1.73·17-s + 0.437·18-s + 2.62·19-s + 21-s − 0.437·22-s − 8.28·23-s + 1.66·24-s + 0.0112·26-s − 27-s + 1.80·28-s − 4.93·29-s + 1.90·31-s + 4.59·32-s + 33-s + 0.760·34-s + ⋯ |
L(s) = 1 | + 0.309·2-s − 0.577·3-s − 0.904·4-s − 0.178·6-s − 0.377·7-s − 0.588·8-s + 0.333·9-s − 0.301·11-s + 0.522·12-s + 0.00715·13-s − 0.116·14-s + 0.722·16-s + 0.421·17-s + 0.103·18-s + 0.603·19-s + 0.218·21-s − 0.0932·22-s − 1.72·23-s + 0.340·24-s + 0.00221·26-s − 0.192·27-s + 0.341·28-s − 0.915·29-s + 0.342·31-s + 0.812·32-s + 0.174·33-s + 0.130·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8380780663\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8380780663\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 0.437T + 2T^{2} \) |
| 13 | \( 1 - 0.0257T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 - 2.62T + 19T^{2} \) |
| 23 | \( 1 + 8.28T + 23T^{2} \) |
| 29 | \( 1 + 4.93T + 29T^{2} \) |
| 31 | \( 1 - 1.90T + 31T^{2} \) |
| 37 | \( 1 - 3.18T + 37T^{2} \) |
| 41 | \( 1 + 2.07T + 41T^{2} \) |
| 43 | \( 1 - 3.85T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 7.00T + 59T^{2} \) |
| 61 | \( 1 + 1.71T + 61T^{2} \) |
| 67 | \( 1 + 7.39T + 67T^{2} \) |
| 71 | \( 1 - 2.36T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 8.85T + 79T^{2} \) |
| 83 | \( 1 - 9.51T + 83T^{2} \) |
| 89 | \( 1 - 9.48T + 89T^{2} \) |
| 97 | \( 1 - 0.755T + 97T^{2} \) |
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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
−7.949104011463636451771446220983, −7.58306809419975974461312449893, −6.35908036888937810188025507040, −5.96614223480607200849140371080, −5.19608784036720373658324623312, −4.56164166418162232038576965125, −3.74989421675935000627548069674, −3.07116058125135429446806901517, −1.75519574771814090094531913994, −0.47742633910306109304189036385,
0.47742633910306109304189036385, 1.75519574771814090094531913994, 3.07116058125135429446806901517, 3.74989421675935000627548069674, 4.56164166418162232038576965125, 5.19608784036720373658324623312, 5.96614223480607200849140371080, 6.35908036888937810188025507040, 7.58306809419975974461312449893, 7.949104011463636451771446220983