L(s) = 1 | + 0.540·2-s + 3-s − 1.70·4-s + 3.09·5-s + 0.540·6-s − 3.94·7-s − 2.00·8-s + 9-s + 1.67·10-s − 2.82·11-s − 1.70·12-s + 0.947·13-s − 2.13·14-s + 3.09·15-s + 2.33·16-s + 17-s + 0.540·18-s + 5.30·19-s − 5.28·20-s − 3.94·21-s − 1.52·22-s − 4.97·23-s − 2.00·24-s + 4.56·25-s + 0.511·26-s + 27-s + 6.74·28-s + ⋯ |
L(s) = 1 | + 0.381·2-s + 0.577·3-s − 0.854·4-s + 1.38·5-s + 0.220·6-s − 1.49·7-s − 0.708·8-s + 0.333·9-s + 0.528·10-s − 0.852·11-s − 0.493·12-s + 0.262·13-s − 0.570·14-s + 0.798·15-s + 0.583·16-s + 0.242·17-s + 0.127·18-s + 1.21·19-s − 1.18·20-s − 0.861·21-s − 0.325·22-s − 1.03·23-s − 0.408·24-s + 0.912·25-s + 0.100·26-s + 0.192·27-s + 1.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 - 0.540T + 2T^{2} \) |
| 5 | \( 1 - 3.09T + 5T^{2} \) |
| 7 | \( 1 + 3.94T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 0.947T + 13T^{2} \) |
| 19 | \( 1 - 5.30T + 19T^{2} \) |
| 23 | \( 1 + 4.97T + 23T^{2} \) |
| 29 | \( 1 + 8.26T + 29T^{2} \) |
| 31 | \( 1 - 1.06T + 31T^{2} \) |
| 37 | \( 1 - 3.33T + 37T^{2} \) |
| 41 | \( 1 - 4.10T + 41T^{2} \) |
| 43 | \( 1 + 6.71T + 43T^{2} \) |
| 47 | \( 1 + 6.96T + 47T^{2} \) |
| 53 | \( 1 + 0.458T + 53T^{2} \) |
| 59 | \( 1 + 5.42T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 0.131T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.139853330987915052669483109630, −7.38116881478017243938376975247, −6.31390242680023724415908080666, −5.81178133303096130494766817515, −5.25192938583273631895769135652, −4.16689408189632201457241973673, −3.25032401089756883227716894864, −2.79179344280112695031034154552, −1.56781329609723102400711326268, 0,
1.56781329609723102400711326268, 2.79179344280112695031034154552, 3.25032401089756883227716894864, 4.16689408189632201457241973673, 5.25192938583273631895769135652, 5.81178133303096130494766817515, 6.31390242680023724415908080666, 7.38116881478017243938376975247, 8.139853330987915052669483109630