L(s) = 1 | + (0.965 − 0.258i)3-s + (0.707 − 0.707i)5-s + (−0.866 − 0.5i)7-s + (0.866 − 0.499i)9-s + (−0.258 + 0.448i)11-s + (0.500 − 0.866i)15-s + (−0.965 − 0.258i)21-s − 1.00i·25-s + (0.707 − 0.707i)27-s − 1.93i·29-s + (0.866 + 0.5i)31-s + (−0.133 + 0.5i)33-s + (−0.965 + 0.258i)35-s + (0.258 − 0.965i)45-s + (0.499 + 0.866i)49-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)3-s + (0.707 − 0.707i)5-s + (−0.866 − 0.5i)7-s + (0.866 − 0.499i)9-s + (−0.258 + 0.448i)11-s + (0.500 − 0.866i)15-s + (−0.965 − 0.258i)21-s − 1.00i·25-s + (0.707 − 0.707i)27-s − 1.93i·29-s + (0.866 + 0.5i)31-s + (−0.133 + 0.5i)33-s + (−0.965 + 0.258i)35-s + (0.258 − 0.965i)45-s + (0.499 + 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.834112381\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.834112381\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.965 + 0.258i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
good | 11 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + 1.93iT - T^{2} \) |
| 31 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (1.36 - 1.36i)T - iT^{2} \) |
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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
−8.693133925173659941477725347068, −7.995484566798799383999532351670, −7.24514250717387780819649159225, −6.47999597715514831658959505385, −5.78674935543002203073117296963, −4.61592543220708469023535626265, −4.02013897291859275797645096920, −2.91834510710367477484202101888, −2.18413776294159165900160428085, −1.01600323489916213778506910091,
1.67449521619124671966554640721, 2.77491066336319699064853449916, 3.08810853831929555066588952458, 4.07546019643860040334203118626, 5.24851897540824668306669792917, 5.93305934399877389395708510540, 6.85469415043407686386222653016, 7.30626521303156206695966616818, 8.510699739053792274624796977600, 8.815475919870389617307887544262