L(s) = 1 | + (1 − i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (0.578 + 4.96i)5-s + 2.44·6-s + (−1.87 + 1.87i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (5.54 + 4.38i)10-s + 19.5·11-s + (2.44 − 2.44i)12-s + (8.03 + 8.03i)13-s + 3.74i·14-s + (−5.37 + 6.79i)15-s − 4·16-s + (−2.19 + 2.19i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (0.115 + 0.993i)5-s + 0.408·6-s + (−0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (0.554 + 0.438i)10-s + 1.77·11-s + (0.204 − 0.204i)12-s + (0.617 + 0.617i)13-s + 0.267i·14-s + (−0.358 + 0.452i)15-s − 0.250·16-s + (−0.129 + 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.31029 + 0.405878i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.31029 + 0.405878i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 + (-0.578 - 4.96i)T \) |
| 7 | \( 1 + (1.87 - 1.87i)T \) |
good | 11 | \( 1 - 19.5T + 121T^{2} \) |
| 13 | \( 1 + (-8.03 - 8.03i)T + 169iT^{2} \) |
| 17 | \( 1 + (2.19 - 2.19i)T - 289iT^{2} \) |
| 19 | \( 1 - 8.25iT - 361T^{2} \) |
| 23 | \( 1 + (17.9 + 17.9i)T + 529iT^{2} \) |
| 29 | \( 1 + 19.7iT - 841T^{2} \) |
| 31 | \( 1 - 30.0T + 961T^{2} \) |
| 37 | \( 1 + (-37.2 + 37.2i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 80.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + (13.6 + 13.6i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (8.17 - 8.17i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (38.8 + 38.8i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 74.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 97.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + (67.1 - 67.1i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 13.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (48.2 + 48.2i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 40.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-34.4 - 34.4i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 157. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (73.2 - 73.2i)T - 9.40e3iT^{2} \) |
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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
−11.93216249400547279021385790158, −11.39939556639010205586175158112, −10.23152131503036682459738385113, −9.510154814323968787565926525110, −8.423840802330755137620479920821, −6.73435492947436244913914947506, −6.08499204668347701034793512382, −4.24678752068275589752388544941, −3.44562576555606804460360603355, −1.98634681396801574477360899676,
1.30810384807650183276600907175, 3.43449695687363813033309195095, 4.53720965940148724811634649469, 5.95989266738741083487321475485, 6.83790128029680957912754095308, 8.117233713559677972262168561728, 8.892053659110617992929470282790, 9.836922339080145240248891002331, 11.58250416167987071241675224127, 12.21222447466871517045853369961