L(s) = 1 | + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (−0.587 + 0.190i)13-s + (0.309 + 0.951i)16-s + (−0.951 − 1.30i)17-s + 0.999i·18-s + 0.618·26-s + (−1.30 − 0.951i)29-s − i·32-s + (0.499 + 1.53i)34-s + (0.309 − 0.951i)36-s + (−1.53 + 0.5i)37-s + (0.190 + 0.587i)41-s + ⋯ |
L(s) = 1 | + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (−0.587 + 0.190i)13-s + (0.309 + 0.951i)16-s + (−0.951 − 1.30i)17-s + 0.999i·18-s + 0.618·26-s + (−1.30 − 0.951i)29-s − i·32-s + (0.499 + 1.53i)34-s + (0.309 − 0.951i)36-s + (−1.53 + 0.5i)37-s + (0.190 + 0.587i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3721493858\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3721493858\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 + 0.309i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
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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
−8.937854603971365743007000525474, −8.194087682415182038248532710161, −7.24345884711880690043258695895, −6.77647710459961808110992484709, −5.90550297720124569487612395058, −4.77413799802218622767226746387, −3.68240626458517311079111153570, −2.82982317526351385012324958370, −1.85697905770701417153395102395, −0.31092466616925168089455889371,
1.70052029853804210968613347936, 2.42315183133040485919879929671, 3.69611484380375774875207772922, 4.98435021692167752823488454390, 5.60135395759732927501750367208, 6.52274568772857517739865075795, 7.29001726624829708285607977023, 7.935967718288424748632669553073, 8.678654596774608049580928189359, 9.202563327004416318755911528161