| L(s) = 1 | − 2.52·3-s − i·5-s + (−2.52 + 0.792i)7-s + 3.37·9-s − 2.52i·11-s + 0.372i·13-s + 2.52i·15-s + 2.37i·17-s + 3.46·19-s + (6.37 − 2i)21-s + 8.51i·23-s − 25-s − 0.939·27-s + 0.372·29-s + 1.87·31-s + ⋯ |
| L(s) = 1 | − 1.45·3-s − 0.447i·5-s + (−0.954 + 0.299i)7-s + 1.12·9-s − 0.761i·11-s + 0.103i·13-s + 0.651i·15-s + 0.575i·17-s + 0.794·19-s + (1.39 − 0.436i)21-s + 1.77i·23-s − 0.200·25-s − 0.180·27-s + 0.0691·29-s + 0.337·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.586000 + 0.257388i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.586000 + 0.257388i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (2.52 - 0.792i)T \) |
| good | 3 | \( 1 + 2.52T + 3T^{2} \) |
| 11 | \( 1 + 2.52iT - 11T^{2} \) |
| 13 | \( 1 - 0.372iT - 13T^{2} \) |
| 17 | \( 1 - 2.37iT - 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 - 8.51iT - 23T^{2} \) |
| 29 | \( 1 - 0.372T + 29T^{2} \) |
| 31 | \( 1 - 1.87T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 8.74iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 7.86T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 6.63T + 59T^{2} \) |
| 61 | \( 1 - 10.7iT - 61T^{2} \) |
| 67 | \( 1 + 6.63iT - 67T^{2} \) |
| 71 | \( 1 + 6.63iT - 71T^{2} \) |
| 73 | \( 1 - 8.74iT - 73T^{2} \) |
| 79 | \( 1 - 15.7iT - 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 5.48iT - 89T^{2} \) |
| 97 | \( 1 + 15.1iT - 97T^{2} \) |
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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
−11.19065460009341437817929747737, −9.971051667194317115781063535253, −9.409717442581509834187196010734, −8.195162254537998051384950082778, −7.04979151966792867516880337073, −5.93005639997966948870183654615, −5.71597763733470785571672356035, −4.44987638897130077156600908382, −3.14816955757727334232730941838, −1.05621862016658153658296402149,
0.55825608469145337280024044786, 2.68044735643863903310863091468, 4.15259080085870851620027122475, 5.15643332215898870662798920037, 6.20235120762582622534961193796, 6.77869463103287199640834252486, 7.62530361432128420885265929318, 9.172939562772565788821727598202, 10.08686533608048375699895617444, 10.59909638239439420925765422506
