L(s) = 1 | + (0.826 + 0.563i)2-s + (−0.733 − 0.680i)3-s + (0.365 + 0.930i)4-s + (0.955 − 0.294i)5-s + (−0.222 − 0.974i)6-s + (−0.222 + 0.974i)8-s + (0.0747 + 0.997i)9-s + (0.955 + 0.294i)10-s + (0.0747 − 0.997i)11-s + (0.365 − 0.930i)12-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)15-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯ |
L(s) = 1 | + (0.826 + 0.563i)2-s + (−0.733 − 0.680i)3-s + (0.365 + 0.930i)4-s + (0.955 − 0.294i)5-s + (−0.222 − 0.974i)6-s + (−0.222 + 0.974i)8-s + (0.0747 + 0.997i)9-s + (0.955 + 0.294i)10-s + (0.0747 − 0.997i)11-s + (0.365 − 0.930i)12-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)15-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.122815745 + 0.1938615830i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.122815745 + 0.1938615830i\) |
\(L(1)\) |
\(\approx\) |
\(1.280466911 + 0.1806316226i\) |
\(L(1)\) |
\(\approx\) |
\(1.280466911 + 0.1806316226i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 + (0.826 + 0.563i)T \) |
| 3 | \( 1 + (-0.733 - 0.680i)T \) |
| 5 | \( 1 + (0.955 - 0.294i)T \) |
| 11 | \( 1 + (0.0747 - 0.997i)T \) |
| 13 | \( 1 + (-0.900 + 0.433i)T \) |
| 17 | \( 1 + (-0.988 + 0.149i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.988 - 0.149i)T \) |
| 29 | \( 1 + (0.623 + 0.781i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.365 - 0.930i)T \) |
| 41 | \( 1 + (-0.222 + 0.974i)T \) |
| 43 | \( 1 + (-0.222 - 0.974i)T \) |
| 47 | \( 1 + (0.826 + 0.563i)T \) |
| 53 | \( 1 + (0.365 + 0.930i)T \) |
| 59 | \( 1 + (0.955 + 0.294i)T \) |
| 61 | \( 1 + (0.365 - 0.930i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.623 - 0.781i)T \) |
| 73 | \( 1 + (0.826 - 0.563i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.900 - 0.433i)T \) |
| 89 | \( 1 + (0.0747 + 0.997i)T \) |
| 97 | \( 1 + T \) |
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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
−33.52325351169627776602221447245, −32.74631799844587404996258560454, −31.61467828956052643899433273280, −30.12109595893963834065689889641, −29.219700139528940594552965133898, −28.368839807944114467393672477788, −27.15709383241467038400905555058, −25.54698009659237326018041340098, −24.20598491182213247613786569680, −22.74698673172678379862502231044, −22.176714330660254953078481240760, −21.109587981789763952545930366350, −20.065815936928411652954614701835, −18.22384584387758560590029146188, −17.10766479957415847740792639573, −15.43997528020018259022189405218, −14.480764046614879481990767539, −12.947518065591294161054323178972, −11.76618446952053708685270123667, −10.318736691759783264319546253761, −9.69962263339120392198709477488, −6.65890100171355105377608803012, −5.47640029502383725092379230024, −4.2263532877540372808585200766, −2.24339800929317109496992518107,
2.29455115051436058876020397693, 4.75403887340930169710031481621, 5.956319884684112567375983290985, 6.95438298346737261907270838384, 8.693319548284248728291819716807, 10.87798358591115479113488115584, 12.2605184342386236118323180501, 13.31569894688231298831225380389, 14.23381942366225381251726984000, 16.11921480034142001995943515043, 17.093602146792523730652118848442, 18.007496912436223229873483196186, 19.80789257991871975676818409348, 21.72961554918514272153972861333, 21.988045959991187279434883995742, 23.71580821367981595569932653160, 24.36005746307786329279093367574, 25.337739661909230008336026520459, 26.72273894166387383440253122905, 28.61335759319650900622284794465, 29.474373037088049719356341633936, 30.33800838877746614975288374930, 31.76872881951831081908138836272, 32.84547753665400128967984795879, 33.91181740593369290783369780544