L(s) = 1 | + 2-s + 4-s + 3.46·5-s + 4.46·7-s + 8-s + 3.46·10-s + 4.73·11-s − 1.46·13-s + 4.46·14-s + 16-s − 3.26·17-s + 1.46·19-s + 3.46·20-s + 4.73·22-s − 7.73·23-s + 6.99·25-s − 1.46·26-s + 4.46·28-s − 1.73·29-s + 31-s + 32-s − 3.26·34-s + 15.4·35-s − 0.732·37-s + 1.46·38-s + 3.46·40-s + 9.19·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.54·5-s + 1.68·7-s + 0.353·8-s + 1.09·10-s + 1.42·11-s − 0.406·13-s + 1.19·14-s + 0.250·16-s − 0.792·17-s + 0.335·19-s + 0.774·20-s + 1.00·22-s − 1.61·23-s + 1.39·25-s − 0.287·26-s + 0.843·28-s − 0.321·29-s + 0.179·31-s + 0.176·32-s − 0.560·34-s + 2.61·35-s − 0.120·37-s + 0.237·38-s + 0.547·40-s + 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.150045430\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.150045430\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 - 4.46T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 3.26T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 + 7.73T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 0.732T + 37T^{2} \) |
| 41 | \( 1 - 9.19T + 41T^{2} \) |
| 43 | \( 1 + 4.92T + 43T^{2} \) |
| 47 | \( 1 + 0.535T + 47T^{2} \) |
| 53 | \( 1 + 9.73T + 53T^{2} \) |
| 59 | \( 1 - 6.73T + 59T^{2} \) |
| 61 | \( 1 + 0.732T + 61T^{2} \) |
| 67 | \( 1 - 4.19T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 5.46T + 73T^{2} \) |
| 79 | \( 1 + 9.26T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 7.73T + 89T^{2} \) |
| 97 | \( 1 - 7.26T + 97T^{2} \) |
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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
−7.72720891319110416624854034773, −7.00072880580872996460561852682, −6.13496630048480084751671968076, −5.87259821424477526384663648364, −4.91920035544676230131853337357, −4.51806151196447967538312361411, −3.67279716365511574786775073178, −2.38077471860875336811629424598, −1.89507983287568893425300357974, −1.26423001087736680828892328610,
1.26423001087736680828892328610, 1.89507983287568893425300357974, 2.38077471860875336811629424598, 3.67279716365511574786775073178, 4.51806151196447967538312361411, 4.91920035544676230131853337357, 5.87259821424477526384663648364, 6.13496630048480084751671968076, 7.00072880580872996460561852682, 7.72720891319110416624854034773