L(s) = 1 | + 2.41·2-s − 3-s + 3.82·4-s − 2.41·6-s + 2.82·7-s + 4.41·8-s + 9-s − 2·11-s − 3.82·12-s + 13-s + 6.82·14-s + 2.99·16-s + 3.65·17-s + 2.41·18-s + 2.82·19-s − 2.82·21-s − 4.82·22-s + 4·23-s − 4.41·24-s + 2.41·26-s − 27-s + 10.8·28-s + 2·29-s − 6.82·31-s − 1.58·32-s + 2·33-s + 8.82·34-s + ⋯ |
L(s) = 1 | + 1.70·2-s − 0.577·3-s + 1.91·4-s − 0.985·6-s + 1.06·7-s + 1.56·8-s + 0.333·9-s − 0.603·11-s − 1.10·12-s + 0.277·13-s + 1.82·14-s + 0.749·16-s + 0.886·17-s + 0.569·18-s + 0.648·19-s − 0.617·21-s − 1.02·22-s + 0.834·23-s − 0.901·24-s + 0.473·26-s − 0.192·27-s + 2.04·28-s + 0.371·29-s − 1.22·31-s − 0.280·32-s + 0.348·33-s + 1.51·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.965583632\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.965583632\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 - 0.343T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 7.65T + 83T^{2} \) |
| 89 | \( 1 - 9.17T + 89T^{2} \) |
| 97 | \( 1 - 7.65T + 97T^{2} \) |
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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
−10.57544344672881670812524448208, −9.247791847062834820436490241771, −7.937311990682410143647630397383, −7.26144302654767755201389179922, −6.23474968480631490897628219316, −5.29061322690231831136106237704, −5.02105349561088502856074001176, −3.92375760443503081123033791234, −2.89805878845605938730540185200, −1.53262660956041559572552533062,
1.53262660956041559572552533062, 2.89805878845605938730540185200, 3.92375760443503081123033791234, 5.02105349561088502856074001176, 5.29061322690231831136106237704, 6.23474968480631490897628219316, 7.26144302654767755201389179922, 7.937311990682410143647630397383, 9.247791847062834820436490241771, 10.57544344672881670812524448208