L(s) = 1 | + (0.540 + 0.841i)2-s + (−0.415 + 0.909i)4-s + (0.281 − 0.959i)7-s + (−0.989 + 0.142i)8-s + (0.909 − 0.415i)9-s + (0.424 − 0.0303i)11-s + (0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (0.841 + 0.540i)18-s + (0.254 + 0.340i)22-s + (0.415 − 0.909i)23-s + (0.755 − 0.654i)25-s + (0.755 + 0.654i)28-s + (0.114 + 0.0855i)29-s + (0.281 − 0.959i)32-s + ⋯ |
L(s) = 1 | + (0.540 + 0.841i)2-s + (−0.415 + 0.909i)4-s + (0.281 − 0.959i)7-s + (−0.989 + 0.142i)8-s + (0.909 − 0.415i)9-s + (0.424 − 0.0303i)11-s + (0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (0.841 + 0.540i)18-s + (0.254 + 0.340i)22-s + (0.415 − 0.909i)23-s + (0.755 − 0.654i)25-s + (0.755 + 0.654i)28-s + (0.114 + 0.0855i)29-s + (0.281 − 0.959i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.693834632\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.693834632\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.540 - 0.841i)T \) |
| 7 | \( 1 + (-0.281 + 0.959i)T \) |
| 23 | \( 1 + (-0.415 + 0.909i)T \) |
good | 3 | \( 1 + (-0.909 + 0.415i)T^{2} \) |
| 5 | \( 1 + (-0.755 + 0.654i)T^{2} \) |
| 11 | \( 1 + (-0.424 + 0.0303i)T + (0.989 - 0.142i)T^{2} \) |
| 13 | \( 1 + (0.540 + 0.841i)T^{2} \) |
| 17 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 19 | \( 1 + (0.281 - 0.959i)T^{2} \) |
| 29 | \( 1 + (-0.114 - 0.0855i)T + (0.281 + 0.959i)T^{2} \) |
| 31 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 37 | \( 1 + (0.613 - 1.64i)T + (-0.755 - 0.654i)T^{2} \) |
| 41 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 43 | \( 1 + (0.148 + 0.682i)T + (-0.909 + 0.415i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.956 - 1.75i)T + (-0.540 + 0.841i)T^{2} \) |
| 59 | \( 1 + (-0.540 - 0.841i)T^{2} \) |
| 61 | \( 1 + (0.909 + 0.415i)T^{2} \) |
| 67 | \( 1 + (-1.59 - 0.114i)T + (0.989 + 0.142i)T^{2} \) |
| 71 | \( 1 + (1.27 - 1.10i)T + (0.142 - 0.989i)T^{2} \) |
| 73 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 79 | \( 1 + (1.45 - 0.425i)T + (0.841 - 0.540i)T^{2} \) |
| 83 | \( 1 + (-0.755 - 0.654i)T^{2} \) |
| 89 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 97 | \( 1 + (0.654 + 0.755i)T^{2} \) |
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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
−8.880439179722998271468587738013, −8.337768323458947980949449257573, −7.36268793134195793752791384265, −6.87026015916381100846818447698, −6.32206055396947344377310835368, −5.14673333475432701946126538693, −4.40431638588042853183159202785, −3.88292872693838331303540971582, −2.80950633526737324764969562516, −1.14801297798876593641822922687,
1.40588953230368602781973875218, 2.19945962397875072550809893425, 3.26593696440466581672752654376, 4.13559650625132308887415303263, 5.05998593269582438881481851790, 5.54030325719669621521179575009, 6.56898805734176205473896209601, 7.37715233212014227246275762534, 8.459499262107384748735371742704, 9.183845001676448245229427200675