L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)6-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 12-s + (−0.309 − 0.951i)13-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + (−0.809 + 0.587i)18-s + (−0.809 − 0.587i)19-s − 23-s + (0.309 + 0.951i)24-s + (0.809 − 0.587i)26-s + (0.309 − 0.951i)27-s + ⋯ |
L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)6-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 12-s + (−0.309 − 0.951i)13-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + (−0.809 + 0.587i)18-s + (−0.809 − 0.587i)19-s − 23-s + (0.309 + 0.951i)24-s + (0.809 − 0.587i)26-s + (0.309 − 0.951i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4991873235 - 0.2766128210i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4991873235 - 0.2766128210i\) |
\(L(1)\) |
\(\approx\) |
\(0.7115302230 + 0.1211801429i\) |
\(L(1)\) |
\(\approx\) |
\(0.7115302230 + 0.1211801429i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 13 | \( 1 + (-0.309 - 0.951i)T \) |
| 17 | \( 1 + (-0.309 + 0.951i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
| 37 | \( 1 + (0.809 - 0.587i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.809 - 0.587i)T \) |
| 53 | \( 1 + (-0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.809 - 0.587i)T \) |
| 61 | \( 1 + (0.309 - 0.951i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.309 - 0.951i)T \) |
| 83 | \( 1 + (-0.309 + 0.951i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.19211194576848793706457765943, −23.53388061722610129626443685773, −22.73792468456177417509282206195, −21.89892451873577512348721951890, −21.34648260674800095848158550479, −20.468110516003255853657398223743, −19.55170773337861226382172998133, −18.47090323001265793131933805587, −17.816835872678331372248834296322, −16.733335212483641120280572588089, −15.869795910058161348971669331694, −14.699106699118261689589366085098, −13.95167416418249128594865590884, −12.68989673883424607719701762607, −11.93295518841234241851159421886, −11.23097180185303978358561281914, −10.27211592114099158887829170670, −9.55978514436661159781887173511, −8.571264812677002907716183973973, −6.81892621077621199843015335522, −5.817756951765343599571797468752, −4.71724297704333465753091566361, −4.10225588674668528316518791027, −2.79096604878814452044871715299, −1.38523644023379621113564749382,
0.356389780231627068750185043235, 2.26422708389347424198495952657, 3.925367469578848677917121240639, 4.96846504746079680460609977510, 5.94320405517603140961103246690, 6.59570064378354611188821488709, 7.73070329918798397645677991472, 8.36958083079085243092359639836, 9.82702436970958976582998290069, 10.88889982255689910327790622573, 12.0955747146649008763049709628, 12.86207049293864471825835830589, 13.51606117376771879870241757714, 14.71913973709518709345989867057, 15.56309348874438972855189025997, 16.47866694962027540554475292552, 17.43855484779923442630683853201, 17.77016133375385728770976403474, 18.87186800350656935900415305142, 19.79957237971929294568810164683, 21.293700185415217733966834873482, 22.13027310944719687309945935657, 22.71530941600994770761541098420, 23.73498189453830948159131438104, 24.13327794281106423975691451735