L(s) = 1 | + (1.13 − 0.847i)2-s + (1.71 + 0.242i)3-s + (0.562 − 1.91i)4-s + (2.14 − 1.17i)6-s + 3.08·7-s + (−0.990 − 2.64i)8-s + (2.88 + 0.831i)9-s + 2.54i·11-s + (1.42 − 3.15i)12-s − 5.06·13-s + (3.49 − 2.61i)14-s + (−3.36 − 2.15i)16-s + 0.214·17-s + (3.96 − 1.50i)18-s − 2.60·19-s + ⋯ |
L(s) = 1 | + (0.800 − 0.599i)2-s + (0.990 + 0.139i)3-s + (0.281 − 0.959i)4-s + (0.876 − 0.481i)6-s + 1.16·7-s + (−0.350 − 0.936i)8-s + (0.960 + 0.277i)9-s + 0.767i·11-s + (0.412 − 0.910i)12-s − 1.40·13-s + (0.934 − 0.700i)14-s + (−0.841 − 0.539i)16-s + 0.0519·17-s + (0.935 − 0.354i)18-s − 0.598·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.95788 - 1.41022i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.95788 - 1.41022i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.13 + 0.847i)T \) |
| 3 | \( 1 + (-1.71 - 0.242i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.08T + 7T^{2} \) |
| 11 | \( 1 - 2.54iT - 11T^{2} \) |
| 13 | \( 1 + 5.06T + 13T^{2} \) |
| 17 | \( 1 - 0.214T + 17T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 + 4.58iT - 31T^{2} \) |
| 37 | \( 1 - 7.67T + 37T^{2} \) |
| 41 | \( 1 + 9.26iT - 41T^{2} \) |
| 43 | \( 1 + 11.4iT - 43T^{2} \) |
| 47 | \( 1 - 10.5iT - 47T^{2} \) |
| 53 | \( 1 - 9.51iT - 53T^{2} \) |
| 59 | \( 1 - 0.428iT - 59T^{2} \) |
| 61 | \( 1 + 1.11iT - 61T^{2} \) |
| 67 | \( 1 - 2.35iT - 67T^{2} \) |
| 71 | \( 1 + 6.12T + 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 + 11.6iT - 79T^{2} \) |
| 83 | \( 1 + 2.29T + 83T^{2} \) |
| 89 | \( 1 - 12.4iT - 89T^{2} \) |
| 97 | \( 1 + 8.04iT - 97T^{2} \) |
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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
−10.57535210972574324741413073612, −9.718491212971493741132212733722, −9.064760428252794389967670928544, −7.67814955726159780520491079621, −7.23869154977527661960626360992, −5.60074929730702588092742422995, −4.64776624220082593221965136550, −3.97436256657372518319782392027, −2.51179051316794439019605599390, −1.78083745729946579777175849418,
2.04025822815776875677402991838, 3.08855010760587810505543450884, 4.34507274486185087246700259328, 5.03523719729153138875607211763, 6.34126299455281692095905729039, 7.36082749390771287592684380090, 8.047555560265886043896007586730, 8.640067489530021194327891397236, 9.741101580054800672630439597746, 10.99702936437118841340643875211