L(s) = 1 | − 1.20·3-s + 6.13·5-s − 7·7-s − 25.5·9-s + 36.3·11-s − 0.689·13-s − 7.36·15-s − 17·17-s − 10.3·19-s + 8.40·21-s − 13.7·23-s − 87.3·25-s + 63.0·27-s − 178.·29-s − 161.·31-s − 43.6·33-s − 42.9·35-s + 376.·37-s + 0.827·39-s − 34.7·41-s − 35.0·43-s − 156.·45-s − 618.·47-s + 49·49-s + 20.4·51-s − 237.·53-s + 223.·55-s + ⋯ |
L(s) = 1 | − 0.231·3-s + 0.549·5-s − 0.377·7-s − 0.946·9-s + 0.997·11-s − 0.0147·13-s − 0.126·15-s − 0.242·17-s − 0.124·19-s + 0.0873·21-s − 0.124·23-s − 0.698·25-s + 0.449·27-s − 1.14·29-s − 0.935·31-s − 0.230·33-s − 0.207·35-s + 1.67·37-s + 0.00339·39-s − 0.132·41-s − 0.124·43-s − 0.519·45-s − 1.91·47-s + 0.142·49-s + 0.0560·51-s − 0.614·53-s + 0.547·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 17 | \( 1 + 17T \) |
good | 3 | \( 1 + 1.20T + 27T^{2} \) |
| 5 | \( 1 - 6.13T + 125T^{2} \) |
| 11 | \( 1 - 36.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.689T + 2.19e3T^{2} \) |
| 19 | \( 1 + 10.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 13.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 161.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 376.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 34.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 35.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 618.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 237.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 280.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 453.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 970.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 912.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.03e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 799.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 774.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 644.T + 9.12e5T^{2} \) |
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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
−10.01510182685439857812117305944, −9.314974668468432583608270713456, −8.500729464188281461741619694962, −7.28236366672756601237559623345, −6.19814161015826971188643404498, −5.66576278498962040837359084403, −4.27890749200180813779493959080, −3.07045322391499379715524918908, −1.70769272189918039355340889461, 0,
1.70769272189918039355340889461, 3.07045322391499379715524918908, 4.27890749200180813779493959080, 5.66576278498962040837359084403, 6.19814161015826971188643404498, 7.28236366672756601237559623345, 8.500729464188281461741619694962, 9.314974668468432583608270713456, 10.01510182685439857812117305944