L(s) = 1 | + (0.669 + 0.743i)4-s + (0.913 − 0.406i)5-s + (−0.104 + 0.994i)16-s + (0.913 + 0.406i)20-s + (1 − 1.73i)23-s + (0.104 + 0.994i)31-s + (−0.309 + 0.951i)37-s + (0.669 − 0.743i)47-s + (0.913 − 0.406i)49-s + (−0.809 + 0.587i)53-s + (0.669 + 0.743i)59-s + (−0.809 + 0.587i)64-s + (0.5 − 0.866i)67-s + (−0.809 − 0.587i)71-s + (0.309 + 0.951i)80-s + ⋯ |
L(s) = 1 | + (0.669 + 0.743i)4-s + (0.913 − 0.406i)5-s + (−0.104 + 0.994i)16-s + (0.913 + 0.406i)20-s + (1 − 1.73i)23-s + (0.104 + 0.994i)31-s + (−0.309 + 0.951i)37-s + (0.669 − 0.743i)47-s + (0.913 − 0.406i)49-s + (−0.809 + 0.587i)53-s + (0.669 + 0.743i)59-s + (−0.809 + 0.587i)64-s + (0.5 − 0.866i)67-s + (−0.809 − 0.587i)71-s + (0.309 + 0.951i)80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.776717600\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.776717600\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 5 | \( 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2} \) |
| 7 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 13 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 31 | \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
| 37 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.669 + 0.743i)T + (-0.104 - 0.994i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
| 61 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 83 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 + (0.913 + 0.406i)T + (0.669 + 0.743i)T^{2} \) |
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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
−8.699388082269804361110926600673, −8.323657007013009732053419116308, −7.21489410127132588033739771660, −6.72145521095265821614800194987, −5.93601282958261537043166749663, −5.08444374596932051890253466336, −4.22769061232340196672228871881, −3.12147147766126177954214330555, −2.40462589015510938630845445862, −1.38893199992097121535414708850,
1.26441198886815818114745929824, 2.18730030709731793448207395987, 2.94747805296358141152182944609, 4.10783126833188156270979387529, 5.40527024660667245116654807363, 5.63994441762852471488596582666, 6.52246401413924918699582532800, 7.14584997816659254791790244055, 7.86077484009843524441054655260, 9.058413826540719535569956926185