L(s) = 1 | − 0.840·2-s − 3-s − 1.29·4-s − 2.01·5-s + 0.840·6-s − 4.87·7-s + 2.76·8-s + 9-s + 1.69·10-s + 11-s + 1.29·12-s + 4.10·14-s + 2.01·15-s + 0.256·16-s + 2.60·17-s − 0.840·18-s − 4.98·19-s + 2.60·20-s + 4.87·21-s − 0.840·22-s + 4.29·23-s − 2.76·24-s − 0.952·25-s − 27-s + 6.30·28-s − 3.22·29-s − 1.69·30-s + ⋯ |
L(s) = 1 | − 0.594·2-s − 0.577·3-s − 0.646·4-s − 0.899·5-s + 0.343·6-s − 1.84·7-s + 0.979·8-s + 0.333·9-s + 0.535·10-s + 0.301·11-s + 0.373·12-s + 1.09·14-s + 0.519·15-s + 0.0641·16-s + 0.630·17-s − 0.198·18-s − 1.14·19-s + 0.581·20-s + 1.06·21-s − 0.179·22-s + 0.895·23-s − 0.565·24-s − 0.190·25-s − 0.192·27-s + 1.19·28-s − 0.598·29-s − 0.308·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05864595485\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05864595485\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.840T + 2T^{2} \) |
| 5 | \( 1 + 2.01T + 5T^{2} \) |
| 7 | \( 1 + 4.87T + 7T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 + 3.22T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 + 7.08T + 37T^{2} \) |
| 41 | \( 1 + 5.87T + 41T^{2} \) |
| 43 | \( 1 + 8.59T + 43T^{2} \) |
| 47 | \( 1 - 7.80T + 47T^{2} \) |
| 53 | \( 1 + 3.45T + 53T^{2} \) |
| 59 | \( 1 + 4.95T + 59T^{2} \) |
| 61 | \( 1 + 9.00T + 61T^{2} \) |
| 67 | \( 1 + 6.12T + 67T^{2} \) |
| 71 | \( 1 - 2.00T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 6.28T + 79T^{2} \) |
| 83 | \( 1 + 9.11T + 83T^{2} \) |
| 89 | \( 1 - 1.64T + 89T^{2} \) |
| 97 | \( 1 + 6.83T + 97T^{2} \) |
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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
−8.240738078324006388638575650200, −7.33706521162094518247983784677, −6.89310763585925048409772534392, −6.08974052396377199091449531642, −5.30732760093497214626813144582, −4.33619261064221969621525454174, −3.73566328746601264098184152597, −3.05991807400622058910935281619, −1.47382267998806788878282464074, −0.15202429079506754711099024550,
0.15202429079506754711099024550, 1.47382267998806788878282464074, 3.05991807400622058910935281619, 3.73566328746601264098184152597, 4.33619261064221969621525454174, 5.30732760093497214626813144582, 6.08974052396377199091449531642, 6.89310763585925048409772534392, 7.33706521162094518247983784677, 8.240738078324006388638575650200