| L(s) = 1 | − i·3-s + i·5-s − 1.70·7-s − 9-s − 3.79·11-s + 1.61·13-s + 15-s + 2.05i·17-s + 0.518·19-s + 1.70i·21-s + (4.69 + 0.988i)23-s − 25-s + i·27-s − 2.88·29-s − 4.16i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447i·5-s − 0.643·7-s − 0.333·9-s − 1.14·11-s + 0.448·13-s + 0.258·15-s + 0.498i·17-s + 0.119·19-s + 0.371i·21-s + (0.978 + 0.206i)23-s − 0.200·25-s + 0.192i·27-s − 0.535·29-s − 0.748i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7012120353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7012120353\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-4.69 - 0.988i)T \) |
| good | 7 | \( 1 + 1.70T + 7T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 - 2.05iT - 17T^{2} \) |
| 19 | \( 1 - 0.518T + 19T^{2} \) |
| 29 | \( 1 + 2.88T + 29T^{2} \) |
| 31 | \( 1 + 4.16iT - 31T^{2} \) |
| 37 | \( 1 - 7.56iT - 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 4.43T + 43T^{2} \) |
| 47 | \( 1 + 0.599iT - 47T^{2} \) |
| 53 | \( 1 + 2.97iT - 53T^{2} \) |
| 59 | \( 1 + 9.67iT - 59T^{2} \) |
| 61 | \( 1 + 2.26iT - 61T^{2} \) |
| 67 | \( 1 + 9.80T + 67T^{2} \) |
| 71 | \( 1 + 9.51iT - 71T^{2} \) |
| 73 | \( 1 + 7.31T + 73T^{2} \) |
| 79 | \( 1 + 7.33T + 79T^{2} \) |
| 83 | \( 1 + 4.63T + 83T^{2} \) |
| 89 | \( 1 - 16.0iT - 89T^{2} \) |
| 97 | \( 1 + 14.4iT - 97T^{2} \) |
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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
−7.79083885634468304412793346132, −7.24873926581265684089710674785, −6.42830260830697497813237776781, −5.91106376363402449954908786866, −5.13890720553686737797797464704, −4.10722223868612594132114402880, −3.13660477237906565075841827397, −2.62596983795445791402070868499, −1.51047889498605509153204005200, −0.20612302665828512958388195899,
1.01473676445012289818274948931, 2.49273397396889220736744229188, 3.09412521033179061712723455533, 4.05990860021105160021412594879, 4.75772828437662658135250425085, 5.57222184281651218821056346616, 6.00957527396334447315867257676, 7.17165781791024932040965554535, 7.61353389124219595275919073717, 8.624099730840928192221664345824
