L(s) = 1 | + (−1.76 − 0.278i)3-s + (−0.587 − 0.809i)4-s − i·5-s + (2.06 + 0.672i)9-s + (0.809 + 1.58i)12-s + (−0.278 + 1.76i)15-s + (−0.309 + 0.951i)16-s + (−0.809 + 0.587i)20-s + (−0.896 + 1.76i)23-s − 25-s + (−1.86 − 0.951i)27-s + (−1.53 − 1.11i)31-s + (−0.672 − 2.06i)36-s + (0.642 + 1.26i)37-s + (0.672 − 2.06i)45-s + ⋯ |
L(s) = 1 | + (−1.76 − 0.278i)3-s + (−0.587 − 0.809i)4-s − i·5-s + (2.06 + 0.672i)9-s + (0.809 + 1.58i)12-s + (−0.278 + 1.76i)15-s + (−0.309 + 0.951i)16-s + (−0.809 + 0.587i)20-s + (−0.896 + 1.76i)23-s − 25-s + (−1.86 − 0.951i)27-s + (−1.53 − 1.11i)31-s + (−0.672 − 2.06i)36-s + (0.642 + 1.26i)37-s + (0.672 − 2.06i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2719940719\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2719940719\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + iT \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 3 | \( 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 13 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.642 - 1.26i)T + (-0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.221 - 1.39i)T + (-0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (0.896 + 0.142i)T + (0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 + 0.0489i)T + (0.951 - 0.309i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \) |
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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
−9.365832963821753976131400375495, −8.131832107868771189631242694627, −7.43455888837884140597670903594, −6.35044867067574322905935488639, −5.76041909710252552555465214140, −5.35457363288893522483073629050, −4.59080411962624072542293109971, −3.93126777635726748280071990278, −1.77298566980545758515108797825, −1.04110780867021398618988765919,
0.26411117566004261905901214670, 2.20002980760064627390965898479, 3.54113289849722366102691657110, 4.14529240666061129387896587704, 5.03673122931825501008402965849, 5.72712222942955676236818797209, 6.63694046439896241776499602868, 7.03350361238538120480696471053, 7.933405995564450088085115142799, 8.866373662273002724050902294679