| L(s) = 1 | − 4·2-s + 16·4-s − 25·5-s + 62·7-s − 64·8-s + 81·9-s + 100·10-s + 82·11-s + 302·13-s − 248·14-s + 256·16-s − 324·18-s − 718·19-s − 400·20-s − 328·22-s + 382·23-s + 625·25-s − 1.20e3·26-s + 992·28-s − 1.02e3·32-s − 1.55e3·35-s + 1.29e3·36-s − 178·37-s + 2.87e3·38-s + 1.60e3·40-s + 2.72e3·41-s + 1.31e3·44-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 5-s + 1.26·7-s − 8-s + 9-s + 10-s + 0.677·11-s + 1.78·13-s − 1.26·14-s + 16-s − 18-s − 1.98·19-s − 20-s − 0.677·22-s + 0.722·23-s + 25-s − 1.78·26-s + 1.26·28-s − 32-s − 1.26·35-s + 36-s − 0.130·37-s + 1.98·38-s + 40-s + 1.61·41-s + 0.677·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.023392494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.023392494\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| good | 3 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 7 | \( 1 - 62 T + p^{4} T^{2} \) |
| 11 | \( 1 - 82 T + p^{4} T^{2} \) |
| 13 | \( 1 - 302 T + p^{4} T^{2} \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( 1 + 718 T + p^{4} T^{2} \) |
| 23 | \( 1 - 382 T + p^{4} T^{2} \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( 1 + 178 T + p^{4} T^{2} \) |
| 41 | \( 1 - 2722 T + p^{4} T^{2} \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( 1 + 2978 T + p^{4} T^{2} \) |
| 53 | \( 1 - 142 T + p^{4} T^{2} \) |
| 59 | \( 1 + 878 T + p^{4} T^{2} \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 79 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( 1 + 15518 T + p^{4} T^{2} \) |
| 97 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
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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
−15.54346536275983412121612948086, −14.70730012669196630884977867739, −12.73061169882321043276402300398, −11.33675700637765922171807742941, −10.75922244240065128286245012570, −8.835856296302565913991277164935, −7.996098955568951993805620055088, −6.61312896302109683748439442892, −4.12356861056942998375626217552, −1.34928882601782603791351089400,
1.34928882601782603791351089400, 4.12356861056942998375626217552, 6.61312896302109683748439442892, 7.996098955568951993805620055088, 8.835856296302565913991277164935, 10.75922244240065128286245012570, 11.33675700637765922171807742941, 12.73061169882321043276402300398, 14.70730012669196630884977867739, 15.54346536275983412121612948086
