L(s) = 1 | + (−1 + i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s + (−4.16 + 2.76i)5-s + 2.44·6-s + (−1.87 + 1.87i)7-s + (2 + 2i)8-s + 2.99i·9-s + (1.39 − 6.93i)10-s + 17.6·11-s + (−2.44 + 2.44i)12-s + (−12.3 − 12.3i)13-s − 3.74i·14-s + (8.48 + 1.71i)15-s − 4·16-s + (18.7 − 18.7i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (−0.832 + 0.553i)5-s + 0.408·6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.139 − 0.693i)10-s + 1.60·11-s + (−0.204 + 0.204i)12-s + (−0.953 − 0.953i)13-s − 0.267i·14-s + (0.565 + 0.114i)15-s − 0.250·16-s + (1.10 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.708763 - 0.280063i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.708763 - 0.280063i\) |
\(L(2)\) |
|
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 | \( 1 + (4.16 - 2.76i)T \) |
| 7 | \( 1 + (1.87 - 1.87i)T \) |
good | 11 | \( 1 - 17.6T + 121T^{2} \) |
| 13 | \( 1 + (12.3 + 12.3i)T + 169iT^{2} \) |
| 17 | \( 1 + (-18.7 + 18.7i)T - 289iT^{2} \) |
| 19 | \( 1 + 25.5iT - 361T^{2} \) |
| 23 | \( 1 + (5.90 + 5.90i)T + 529iT^{2} \) |
| 29 | \( 1 - 15.3iT - 841T^{2} \) |
| 31 | \( 1 - 11.3T + 961T^{2} \) |
| 37 | \( 1 + (-25.5 + 25.5i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 58.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-0.282 - 0.282i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (48.6 - 48.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-3.93 - 3.93i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 86.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 29.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + (65.7 - 65.7i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 1.19T + 5.04e3T^{2} \) |
| 73 | \( 1 + (87.0 + 87.0i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 55.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (91.8 + 91.8i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 103. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-87.4 + 87.4i)T - 9.40e3iT^{2} \) |
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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
−11.87325796381324009335803837798, −11.19708654102160028428538395112, −9.959994412231403794043385115208, −9.028770980463494872248190739792, −7.69656795572317364578239455273, −7.08275081346513635259276216922, −6.07764550631133461961401559953, −4.69245340970319970037609074049, −2.94037550267987467208196855000, −0.61852412234657013608966523889,
1.31220516719191800821502814560, 3.69218816387769694716025608043, 4.34617969650831578672590800157, 6.08119539071312548819905826570, 7.36450020396851495597017619670, 8.437677552894780164534436890017, 9.508156162748252131720181162485, 10.15740585254561927242564295438, 11.55733417642943913677931304381, 11.94622429212325343105494379978