L(s) = 1 | + (0.933 − 0.358i)2-s + (−0.838 + 0.544i)3-s + (0.743 − 0.669i)4-s + (−0.587 + 0.809i)6-s + (0.333 − 0.942i)7-s + (0.453 − 0.891i)8-s + (0.406 − 0.913i)9-s + (−0.0261 + 0.999i)11-s + (−0.258 + 0.965i)12-s + (−0.793 + 0.608i)13-s + (−0.0261 − 0.999i)14-s + (0.104 − 0.994i)16-s + (0.972 − 0.233i)17-s + (0.0523 − 0.998i)18-s + (0.333 − 0.942i)19-s + ⋯ |
L(s) = 1 | + (0.933 − 0.358i)2-s + (−0.838 + 0.544i)3-s + (0.743 − 0.669i)4-s + (−0.587 + 0.809i)6-s + (0.333 − 0.942i)7-s + (0.453 − 0.891i)8-s + (0.406 − 0.913i)9-s + (−0.0261 + 0.999i)11-s + (−0.258 + 0.965i)12-s + (−0.793 + 0.608i)13-s + (−0.0261 − 0.999i)14-s + (0.104 − 0.994i)16-s + (0.972 − 0.233i)17-s + (0.0523 − 0.998i)18-s + (0.333 − 0.942i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.310026807 - 1.926973703i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.310026807 - 1.926973703i\) |
\(L(1)\) |
\(\approx\) |
\(1.387720236 - 0.4878903063i\) |
\(L(1)\) |
\(\approx\) |
\(1.387720236 - 0.4878903063i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (0.933 - 0.358i)T \) |
| 3 | \( 1 + (-0.838 + 0.544i)T \) |
| 7 | \( 1 + (0.333 - 0.942i)T \) |
| 11 | \( 1 + (-0.0261 + 0.999i)T \) |
| 13 | \( 1 + (-0.793 + 0.608i)T \) |
| 17 | \( 1 + (0.972 - 0.233i)T \) |
| 19 | \( 1 + (0.333 - 0.942i)T \) |
| 23 | \( 1 + (-0.0784 - 0.996i)T \) |
| 29 | \( 1 + (-0.629 - 0.777i)T \) |
| 31 | \( 1 + (0.725 - 0.688i)T \) |
| 37 | \( 1 + (-0.333 + 0.942i)T \) |
| 41 | \( 1 + (0.891 - 0.453i)T \) |
| 43 | \( 1 + (0.760 + 0.649i)T \) |
| 47 | \( 1 + (0.156 + 0.987i)T \) |
| 53 | \( 1 + (0.358 - 0.933i)T \) |
| 59 | \( 1 + (-0.998 + 0.0523i)T \) |
| 61 | \( 1 + (-0.453 - 0.891i)T \) |
| 67 | \( 1 + (-0.933 + 0.358i)T \) |
| 71 | \( 1 + (-0.878 - 0.477i)T \) |
| 73 | \( 1 + (0.923 - 0.382i)T \) |
| 79 | \( 1 + (-0.987 - 0.156i)T \) |
| 83 | \( 1 + (0.978 - 0.207i)T \) |
| 89 | \( 1 + (-0.725 + 0.688i)T \) |
| 97 | \( 1 + (-0.913 + 0.406i)T \) |
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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
−17.9299209018713583896509192415, −16.96418141211662048297417727081, −16.71503883653462169585504166104, −15.83257861997511473566590190559, −15.43054262568331173871807003903, −14.408003036471332037752269560323, −14.12779713762968246760997679951, −13.20729461521509343469227991006, −12.55590456928396853239377459064, −12.072414125370632945568539590865, −11.677422098013711461240241488634, −10.81773379316854402073852616159, −10.29284700111745027715268997063, −9.11825282459257969934947725205, −8.22777516675000918790554454239, −7.64824496782186832472699154567, −7.16551714014139402079276361212, −6.03295540462078258002915224576, −5.63604552796669210795158210392, −5.40057653211624922421771126100, −4.46339309342839595005358151064, −3.44215348894478726098842056719, −2.82011231225308851998241801338, −1.90028812498719889305387748725, −1.138310497467197391860897441904,
0.48249715377818707121239178912, 1.32436225406848720026658719496, 2.27781943536521057890755278916, 3.13925936109697898060986919806, 4.100544116873770947355606917484, 4.625369290621564432217723733924, 4.866440186868152143543335632314, 5.88668244273483777648055749199, 6.5535388457016050545115457185, 7.26728205868099798441974753850, 7.71732656148526330727427714338, 9.28659315933176835884370456917, 9.82356886075781839252689880663, 10.28389416972885227290463066320, 11.062360237772725609591625121067, 11.569009716497973499086875276036, 12.23746337470000278729065873088, 12.71344288399507850924373808634, 13.609480624750443320765033041832, 14.25548177464459392877561780990, 14.877936701127482099205297284435, 15.38659059545560491843535492914, 16.22454615225095341674085028709, 16.77206920766676533497490299824, 17.33315368558453373012704905927