| L(s) = 1 | − 2-s + 4-s − 3·5-s + 7-s − 8-s + 3·10-s + 5·13-s − 14-s + 16-s − 4·19-s − 3·20-s − 3·23-s + 4·25-s − 5·26-s + 28-s − 6·29-s + 2·31-s − 32-s − 3·35-s − 4·37-s + 4·38-s + 3·40-s − 6·41-s − 4·43-s + 3·46-s − 6·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s + 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.670·20-s − 0.625·23-s + 4/5·25-s − 0.980·26-s + 0.188·28-s − 1.11·29-s + 0.359·31-s − 0.176·32-s − 0.507·35-s − 0.657·37-s + 0.648·38-s + 0.474·40-s − 0.937·41-s − 0.609·43-s + 0.442·46-s − 0.875·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6160303319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6160303319\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.94918041874762, −14.73058088144850, −13.69562124313132, −13.46238781734109, −12.61783810439071, −12.10635880210312, −11.67462520136470, −11.05202791337589, −10.89099222422372, −10.18225617284145, −9.549970654022553, −8.691233151037370, −8.501547931724335, −8.021547132534598, −7.474031501480066, −6.830801108884540, −6.308657798487501, −5.609696149081141, −4.847980462998100, −3.988613788674923, −3.758388616941341, −2.989030856494603, −1.977293538848453, −1.365607432768064, −0.3387495611256139,
0.3387495611256139, 1.365607432768064, 1.977293538848453, 2.989030856494603, 3.758388616941341, 3.988613788674923, 4.847980462998100, 5.609696149081141, 6.308657798487501, 6.830801108884540, 7.474031501480066, 8.021547132534598, 8.501547931724335, 8.691233151037370, 9.549970654022553, 10.18225617284145, 10.89099222422372, 11.05202791337589, 11.67462520136470, 12.10635880210312, 12.61783810439071, 13.46238781734109, 13.69562124313132, 14.73058088144850, 14.94918041874762
