| L(s) = 1 | − 3.12·3-s − 5-s − 1.51·7-s + 6.76·9-s + 4.24·11-s + 4.15·13-s + 3.12·15-s − 3.51·17-s − 19-s + 4.73·21-s − 8.73·23-s + 25-s − 11.7·27-s + 1.45·29-s − 4.96·31-s − 13.2·33-s + 1.51·35-s + 7.60·37-s − 12.9·39-s − 9.21·41-s − 8.31·43-s − 6.76·45-s + 5.28·47-s − 4.70·49-s + 10.9·51-s + 0.155·53-s − 4.24·55-s + ⋯ |
| L(s) = 1 | − 1.80·3-s − 0.447·5-s − 0.572·7-s + 2.25·9-s + 1.28·11-s + 1.15·13-s + 0.806·15-s − 0.852·17-s − 0.229·19-s + 1.03·21-s − 1.82·23-s + 0.200·25-s − 2.26·27-s + 0.270·29-s − 0.892·31-s − 2.31·33-s + 0.256·35-s + 1.25·37-s − 2.07·39-s − 1.43·41-s − 1.26·43-s − 1.00·45-s + 0.770·47-s − 0.672·49-s + 1.53·51-s + 0.0213·53-s − 0.573·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 3.12T + 3T^{2} \) |
| 7 | \( 1 + 1.51T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 - 4.15T + 13T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 23 | \( 1 + 8.73T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 + 4.96T + 31T^{2} \) |
| 37 | \( 1 - 7.60T + 37T^{2} \) |
| 41 | \( 1 + 9.21T + 41T^{2} \) |
| 43 | \( 1 + 8.31T + 43T^{2} \) |
| 47 | \( 1 - 5.28T + 47T^{2} \) |
| 53 | \( 1 - 0.155T + 53T^{2} \) |
| 59 | \( 1 + 2.48T + 59T^{2} \) |
| 61 | \( 1 + 4.49T + 61T^{2} \) |
| 67 | \( 1 - 7.43T + 67T^{2} \) |
| 71 | \( 1 - 8.49T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 0.310T + 79T^{2} \) |
| 83 | \( 1 + 8.96T + 83T^{2} \) |
| 89 | \( 1 - 0.719T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.14045193155216073244453957464, −9.240599745448566300270306069503, −8.131579358308458661904477506025, −6.77373143567401309148253712603, −6.43879339739781149747044737983, −5.64800226140176224046869953244, −4.38421611084775338443680824516, −3.76015136093424328614137817959, −1.50297959141069652117903727409, 0,
1.50297959141069652117903727409, 3.76015136093424328614137817959, 4.38421611084775338443680824516, 5.64800226140176224046869953244, 6.43879339739781149747044737983, 6.77373143567401309148253712603, 8.131579358308458661904477506025, 9.240599745448566300270306069503, 10.14045193155216073244453957464
