| L(s) = 1 | − 0.815·2-s − 3.26·3-s − 1.33·4-s + 2.66·6-s − 7-s + 2.71·8-s + 7.64·9-s − 11-s + 4.35·12-s + 4.02·13-s + 0.815·14-s + 0.452·16-s − 7.77·17-s − 6.23·18-s + 6.63·19-s + 3.26·21-s + 0.815·22-s + 1.29·23-s − 8.87·24-s − 3.28·26-s − 15.1·27-s + 1.33·28-s − 6.31·29-s + 6.71·31-s − 5.80·32-s + 3.26·33-s + 6.33·34-s + ⋯ |
| L(s) = 1 | − 0.576·2-s − 1.88·3-s − 0.667·4-s + 1.08·6-s − 0.377·7-s + 0.961·8-s + 2.54·9-s − 0.301·11-s + 1.25·12-s + 1.11·13-s + 0.217·14-s + 0.113·16-s − 1.88·17-s − 1.46·18-s + 1.52·19-s + 0.712·21-s + 0.173·22-s + 0.269·23-s − 1.81·24-s − 0.644·26-s − 2.91·27-s + 0.252·28-s − 1.17·29-s + 1.20·31-s − 1.02·32-s + 0.568·33-s + 1.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3677560075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3677560075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.815T + 2T^{2} \) |
| 3 | \( 1 + 3.26T + 3T^{2} \) |
| 13 | \( 1 - 4.02T + 13T^{2} \) |
| 17 | \( 1 + 7.77T + 17T^{2} \) |
| 19 | \( 1 - 6.63T + 19T^{2} \) |
| 23 | \( 1 - 1.29T + 23T^{2} \) |
| 29 | \( 1 + 6.31T + 29T^{2} \) |
| 31 | \( 1 - 6.71T + 31T^{2} \) |
| 37 | \( 1 + 9.84T + 37T^{2} \) |
| 41 | \( 1 + 4.73T + 41T^{2} \) |
| 43 | \( 1 - 0.544T + 43T^{2} \) |
| 47 | \( 1 - 2.66T + 47T^{2} \) |
| 53 | \( 1 + 3.65T + 53T^{2} \) |
| 59 | \( 1 - 4.02T + 59T^{2} \) |
| 61 | \( 1 + 5.19T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 2.60T + 73T^{2} \) |
| 79 | \( 1 + 8.11T + 79T^{2} \) |
| 83 | \( 1 - 5.75T + 83T^{2} \) |
| 89 | \( 1 - 7.73T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−9.305147370982588664178068766608, −8.586795089564007826183617982732, −7.44634995745889477883586045508, −6.79491371545304877266600796757, −5.98300404419064385536107867933, −5.20685131284372761767797317506, −4.55347404206960497160610918165, −3.61595367165694000668479746687, −1.60248066377051603839330094410, −0.51701981302563167635350925351,
0.51701981302563167635350925351, 1.60248066377051603839330094410, 3.61595367165694000668479746687, 4.55347404206960497160610918165, 5.20685131284372761767797317506, 5.98300404419064385536107867933, 6.79491371545304877266600796757, 7.44634995745889477883586045508, 8.586795089564007826183617982732, 9.305147370982588664178068766608
