L(s) = 1 | − i·2-s + (−0.707 − 0.707i)3-s − 4-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)6-s + i·8-s + i·9-s + (0.707 − 0.707i)10-s + (0.707 − 0.707i)11-s + (0.707 + 0.707i)12-s + 13-s − i·15-s + 16-s + 18-s + i·19-s + (−0.707 − 0.707i)20-s + ⋯ |
L(s) = 1 | − i·2-s + (−0.707 − 0.707i)3-s − 4-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)6-s + i·8-s + i·9-s + (0.707 − 0.707i)10-s + (0.707 − 0.707i)11-s + (0.707 + 0.707i)12-s + 13-s − i·15-s + 16-s + 18-s + i·19-s + (−0.707 − 0.707i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0465 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0465 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.066469631 - 1.017966179i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.066469631 - 1.017966179i\) |
\(L(1)\) |
\(\approx\) |
\(0.8404659796 - 0.5374673593i\) |
\(L(1)\) |
\(\approx\) |
\(0.8404659796 - 0.5374673593i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 11 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 + (0.707 - 0.707i)T \) |
| 29 | \( 1 + (-0.707 - 0.707i)T \) |
| 31 | \( 1 + (-0.707 - 0.707i)T \) |
| 37 | \( 1 + (0.707 + 0.707i)T \) |
| 41 | \( 1 + (0.707 - 0.707i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 - iT \) |
| 61 | \( 1 + (0.707 - 0.707i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.707 + 0.707i)T \) |
| 73 | \( 1 + (0.707 + 0.707i)T \) |
| 79 | \( 1 + (0.707 - 0.707i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (0.707 + 0.707i)T \) |
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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
−28.64052041436982823655320099563, −28.04056687804854903097894452865, −27.16904219248149258265353660437, −25.893117046149125901352625623762, −25.22097127843420203348670698344, −23.97547001767572985064524104329, −23.163835450410543026202007188188, −22.08749195340224838046977675629, −21.304181340290929977875469268112, −20.04673561725111237127504267326, −18.21171388784260202152695717708, −17.45656502779304948950889402518, −16.66848000356350032117414239748, −15.762622746942457547177367091492, −14.73107438311144787378516150315, −13.44373372343915097522536364191, −12.39865344190506494682822716241, −10.828200517957034466265427125414, −9.3989389995519763351271608825, −8.95000652481262482695264314073, −7.04591050183741815853336002780, −5.914855099386449715940827932400, −5.00753774132285936283521415677, −3.90620393584661786679158677310, −1.028427592119508514596104409711,
0.96920045969048869322380202660, 2.21451011697088777873194813022, 3.73733941075828220862983708265, 5.55447996823304017732847671032, 6.44353938097737956140880596542, 8.12925354977670364495999194388, 9.55022063816824055719577028055, 10.85327752319942792056246588287, 11.37802795597352980230999150356, 12.73521229464190250832872625854, 13.61323190702275513947841390927, 14.52150374107314659716874412424, 16.61655532565044400706742325637, 17.51502821024931154992006284961, 18.67813161460662234422070960688, 18.8992602453156359741840638794, 20.509578656241156500198515585323, 21.607683555952599830581238569615, 22.48703979382474311162939262044, 23.150565705732241763443801647400, 24.47970725280634799134514195143, 25.63726567718300948646105282593, 26.88677282860871406256465082334, 27.89883046894129371580915458288, 28.92167060237542181435664995847