L(s) = 1 | + (−0.951 − 0.309i)3-s + (0.809 + 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.587 + 0.809i)13-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (0.587 + 0.809i)23-s + (−0.587 − 0.809i)27-s + (0.309 − 0.951i)29-s + (−0.309 − 0.951i)31-s + (−0.951 + 0.309i)33-s + (−0.587 + 0.809i)37-s + (0.809 − 0.587i)39-s + (0.809 + 0.587i)41-s − i·43-s + ⋯ |
L(s) = 1 | + (−0.951 − 0.309i)3-s + (0.809 + 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.587 + 0.809i)13-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (0.587 + 0.809i)23-s + (−0.587 − 0.809i)27-s + (0.309 − 0.951i)29-s + (−0.309 − 0.951i)31-s + (−0.951 + 0.309i)33-s + (−0.587 + 0.809i)37-s + (0.809 − 0.587i)39-s + (0.809 + 0.587i)41-s − i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.093069759 - 0.3175661213i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.093069759 - 0.3175661213i\) |
\(L(1)\) |
\(\approx\) |
\(0.8595497209 - 0.1097881126i\) |
\(L(1)\) |
\(\approx\) |
\(0.8595497209 - 0.1097881126i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 11 | \( 1 + (0.809 - 0.587i)T \) |
| 13 | \( 1 + (-0.587 + 0.809i)T \) |
| 17 | \( 1 + (0.951 - 0.309i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
| 23 | \( 1 + (0.587 + 0.809i)T \) |
| 29 | \( 1 + (0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
| 37 | \( 1 + (-0.587 + 0.809i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.951 - 0.309i)T \) |
| 53 | \( 1 + (0.951 + 0.309i)T \) |
| 59 | \( 1 + (0.809 + 0.587i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 + (0.951 - 0.309i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.587 - 0.809i)T \) |
| 79 | \( 1 + (-0.309 + 0.951i)T \) |
| 83 | \( 1 + (0.951 - 0.309i)T \) |
| 89 | \( 1 + (-0.809 + 0.587i)T \) |
| 97 | \( 1 + (-0.951 - 0.309i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.97835397207462386922276377613, −20.19303407271438691883306474632, −19.30722063112235837766364336305, −18.4990711865761479634183823877, −17.6584384503632171464127613410, −17.12546755984917462477392974969, −16.44034102702196197433418661396, −15.69343430286823474864333715513, −14.64981337543997142986602936059, −14.38809868230009429314411019321, −12.68476302279130183949177338912, −12.542955861326721838519932170564, −11.74348941037240666956084844338, −10.62941479108255009491738213641, −10.264894369320749318895252404013, −9.384546241101395615908055031170, −8.41503867521044637619125797990, −7.29088071470357713412825987518, −6.69402698580425139282768508448, −5.65644782076122410367220670422, −5.091338939497620796037637554776, −4.087470942048589069841596637351, −3.3049710617652834660409473200, −1.87523079622243170351192274688, −0.85393212484329976258424732308,
0.71438378885426752368954331268, 1.66772114137797101587448616826, 2.84449036476681089849373551981, 4.06007441380048104557557208703, 4.83952454507661257076530977821, 5.73167005652174782483187662473, 6.51690345339153737335104686201, 7.200020722118828785786958213795, 8.06659857547289456555793424680, 9.26732389500345808607668937092, 9.82196583690047499710085626324, 10.93294908611191148764141303467, 11.629016144002987104456717373576, 11.99679198482486678060427049509, 13.10977952605070296948759743327, 13.69727054771210661566491971721, 14.658777310003659640599792013233, 15.49133048765855798028825545855, 16.573252667511982782236092329426, 16.816013989064876607810894524821, 17.6226532541211938436284352443, 18.44249575575454296936594566517, 19.32818259783840341125098265150, 19.55484469926925070183516240474, 21.07048835307896116449635825007