L(s) = 1 | + (0.587 + 0.809i)5-s + (−0.951 + 0.309i)9-s + (−1.76 − 0.896i)13-s + (−0.278 + 1.76i)17-s + (−0.309 + 0.951i)25-s + (0.5 + 0.363i)29-s + (−0.809 + 1.58i)37-s + (0.363 + 1.11i)41-s + (−0.809 − 0.587i)45-s + i·49-s + (−0.309 + 0.0489i)53-s + (−1.53 − 0.5i)61-s + (−0.309 − 1.95i)65-s + (1.76 − 0.896i)73-s + (0.809 − 0.587i)81-s + ⋯ |
L(s) = 1 | + (0.587 + 0.809i)5-s + (−0.951 + 0.309i)9-s + (−1.76 − 0.896i)13-s + (−0.278 + 1.76i)17-s + (−0.309 + 0.951i)25-s + (0.5 + 0.363i)29-s + (−0.809 + 1.58i)37-s + (0.363 + 1.11i)41-s + (−0.809 − 0.587i)45-s + i·49-s + (−0.309 + 0.0489i)53-s + (−1.53 − 0.5i)61-s + (−0.309 − 1.95i)65-s + (1.76 − 0.896i)73-s + (0.809 − 0.587i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8174715178\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8174715178\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.587 - 0.809i)T \) |
good | 3 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (1.76 + 0.896i)T + (0.587 + 0.809i)T^{2} \) |
| 17 | \( 1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 1.58i)T + (-0.587 - 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.0489i)T + (0.951 - 0.309i)T^{2} \) |
| 59 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-1.76 + 0.896i)T + (0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 89 | \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.896 - 0.142i)T + (0.951 - 0.309i)T^{2} \) |
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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
−9.182229580049751600256388493750, −8.146938425186109621599093753143, −7.78508216438129598431138658010, −6.71274017716045485443569827646, −6.14916671452523482012374719017, −5.35604736326213731318133543229, −4.62718449959383427598071017231, −3.27105938648442573324135962377, −2.73289589462564306078027304816, −1.76960363162539352839939916410,
0.44826297439707773287003253848, 2.12193834580992001713722763947, 2.67611476656966156894670040029, 4.00589478731694201348028693452, 5.03134351098376661682499574530, 5.25895097093428642204177716159, 6.34702028093647513727798338385, 7.10609464206165661247332963511, 7.81677272919847641987148596867, 8.939332524590270077754794663965