L(s) = 1 | + (0.996 + 0.0859i)2-s + (0.584 + 0.811i)3-s + (0.985 + 0.171i)4-s + (−0.683 + 0.729i)5-s + (0.512 + 0.858i)6-s + (0.869 − 0.493i)7-s + (0.966 + 0.255i)8-s + (−0.317 + 0.948i)9-s + (−0.744 + 0.668i)10-s + (−0.890 − 0.455i)11-s + (0.436 + 0.899i)12-s + (−0.474 + 0.880i)13-s + (0.908 − 0.417i)14-s + (−0.991 − 0.128i)15-s + (0.941 + 0.337i)16-s + (0.996 + 0.0859i)17-s + ⋯ |
L(s) = 1 | + (0.996 + 0.0859i)2-s + (0.584 + 0.811i)3-s + (0.985 + 0.171i)4-s + (−0.683 + 0.729i)5-s + (0.512 + 0.858i)6-s + (0.869 − 0.493i)7-s + (0.966 + 0.255i)8-s + (−0.317 + 0.948i)9-s + (−0.744 + 0.668i)10-s + (−0.890 − 0.455i)11-s + (0.436 + 0.899i)12-s + (−0.474 + 0.880i)13-s + (0.908 − 0.417i)14-s + (−0.991 − 0.128i)15-s + (0.941 + 0.337i)16-s + (0.996 + 0.0859i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.046516329 + 1.576158490i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.046516329 + 1.576158490i\) |
\(L(1)\) |
\(\approx\) |
\(1.887049313 + 0.8440191746i\) |
\(L(1)\) |
\(\approx\) |
\(1.887049313 + 0.8440191746i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 293 | \( 1 \) |
good | 2 | \( 1 + (0.996 + 0.0859i)T \) |
| 3 | \( 1 + (0.584 + 0.811i)T \) |
| 5 | \( 1 + (-0.683 + 0.729i)T \) |
| 7 | \( 1 + (0.869 - 0.493i)T \) |
| 11 | \( 1 + (-0.890 - 0.455i)T \) |
| 13 | \( 1 + (-0.474 + 0.880i)T \) |
| 17 | \( 1 + (0.996 + 0.0859i)T \) |
| 19 | \( 1 + (-0.991 - 0.128i)T \) |
| 23 | \( 1 + (0.512 - 0.858i)T \) |
| 29 | \( 1 + (0.107 - 0.994i)T \) |
| 31 | \( 1 + (-0.683 - 0.729i)T \) |
| 37 | \( 1 + (0.772 + 0.635i)T \) |
| 41 | \( 1 + (0.0215 + 0.999i)T \) |
| 43 | \( 1 + (0.714 + 0.699i)T \) |
| 47 | \( 1 + (-0.847 + 0.530i)T \) |
| 53 | \( 1 + (-0.890 - 0.455i)T \) |
| 59 | \( 1 + (-0.618 - 0.785i)T \) |
| 61 | \( 1 + (0.357 - 0.933i)T \) |
| 67 | \( 1 + (-0.234 - 0.972i)T \) |
| 71 | \( 1 + (-0.150 + 0.988i)T \) |
| 73 | \( 1 + (0.985 - 0.171i)T \) |
| 79 | \( 1 + (0.357 - 0.933i)T \) |
| 83 | \( 1 + (-0.954 + 0.296i)T \) |
| 89 | \( 1 + (0.357 + 0.933i)T \) |
| 97 | \( 1 + (0.651 - 0.758i)T \) |
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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
−25.239958085738290218235466641615, −24.09010434572816673409426651241, −23.672015523693490224432172911314, −22.92052218201899407767030463419, −21.385376518950134270106537479637, −20.806141999071880001576385416147, −19.99682441601270144057509630312, −19.2185504598945308816248399911, −18.11307682626019618580913308232, −16.99495597842737740490170600391, −15.65644981548331155707331559456, −14.99263891829271811817191579066, −14.24096689770517149068420219863, −12.86219112619087209071636228662, −12.59820147873086486438993598144, −11.695596303919908049507924632421, −10.52735567557791790361064477269, −8.88140160568166385614107236190, −7.77882488487196709738350478666, −7.37294606834497320816063750866, −5.644739566237902457476441983650, −4.9562195864078427425375820088, −3.59685068708239118705639992728, −2.480694892536270377072754997485, −1.34334002557286380325700190028,
2.21125504947857354637513816376, 3.16595034072283208344772527433, 4.25658973977933519699228220693, 4.85923679961303344497292650463, 6.3339712913735405900312259774, 7.70317073175513061801524664624, 8.12068266666796120634654188535, 9.94389556297999140902071283628, 10.97186132791702725284497530556, 11.37141017477130772026571804923, 12.826015080536339365723398091842, 14.009861067446298155119638036709, 14.614467257905438632252448370291, 15.18366884461335411033186636715, 16.27092832327864974744770098936, 16.96710419825761529408166349603, 18.73486184871992276359542278768, 19.486460206638172622260795002358, 20.58876418249612379386131707691, 21.206895901193920223996735841194, 21.89431890501736238556430461778, 23.02292227298479570176787588683, 23.66521527428193786312641888129, 24.54899278589370609883028814805, 25.79072635427855830412862708278