L(s) = 1 | + (0.879 − 0.476i)2-s + (−0.792 + 0.610i)3-s + (0.545 − 0.838i)4-s + (−0.981 − 0.194i)5-s + (−0.405 + 0.914i)6-s + (0.758 − 0.651i)7-s + (0.0797 − 0.996i)8-s + (0.254 − 0.967i)9-s + (−0.954 + 0.297i)10-s + (0.999 + 0.0354i)11-s + (0.0797 + 0.996i)12-s + (0.185 − 0.982i)13-s + (0.355 − 0.934i)14-s + (0.895 − 0.445i)15-s + (−0.405 − 0.914i)16-s + (−0.468 + 0.883i)17-s + ⋯ |
L(s) = 1 | + (0.879 − 0.476i)2-s + (−0.792 + 0.610i)3-s + (0.545 − 0.838i)4-s + (−0.981 − 0.194i)5-s + (−0.405 + 0.914i)6-s + (0.758 − 0.651i)7-s + (0.0797 − 0.996i)8-s + (0.254 − 0.967i)9-s + (−0.954 + 0.297i)10-s + (0.999 + 0.0354i)11-s + (0.0797 + 0.996i)12-s + (0.185 − 0.982i)13-s + (0.355 − 0.934i)14-s + (0.895 − 0.445i)15-s + (−0.405 − 0.914i)16-s + (−0.468 + 0.883i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6861149456 - 1.249554006i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6861149456 - 1.249554006i\) |
\(L(1)\) |
\(\approx\) |
\(1.087721888 - 0.5332941682i\) |
\(L(1)\) |
\(\approx\) |
\(1.087721888 - 0.5332941682i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 709 | \( 1 \) |
good | 2 | \( 1 + (0.879 - 0.476i)T \) |
| 3 | \( 1 + (-0.792 + 0.610i)T \) |
| 5 | \( 1 + (-0.981 - 0.194i)T \) |
| 7 | \( 1 + (0.758 - 0.651i)T \) |
| 11 | \( 1 + (0.999 + 0.0354i)T \) |
| 13 | \( 1 + (0.185 - 0.982i)T \) |
| 17 | \( 1 + (-0.468 + 0.883i)T \) |
| 19 | \( 1 + (-0.903 + 0.429i)T \) |
| 23 | \( 1 + (-0.870 - 0.492i)T \) |
| 29 | \( 1 + (-0.722 - 0.691i)T \) |
| 31 | \( 1 + (-0.0620 + 0.998i)T \) |
| 37 | \( 1 + (0.758 - 0.651i)T \) |
| 41 | \( 1 + (0.842 - 0.537i)T \) |
| 43 | \( 1 + (-0.813 - 0.582i)T \) |
| 47 | \( 1 + (0.861 - 0.507i)T \) |
| 53 | \( 1 + (-0.833 - 0.552i)T \) |
| 59 | \( 1 + (0.0797 - 0.996i)T \) |
| 61 | \( 1 + (-0.560 + 0.828i)T \) |
| 67 | \( 1 + (-0.167 - 0.985i)T \) |
| 71 | \( 1 + (-0.954 - 0.297i)T \) |
| 73 | \( 1 + (0.484 - 0.874i)T \) |
| 79 | \( 1 + (0.684 + 0.728i)T \) |
| 83 | \( 1 + (-0.931 - 0.364i)T \) |
| 89 | \( 1 + (-0.671 - 0.740i)T \) |
| 97 | \( 1 + (0.969 - 0.245i)T \) |
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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
−23.02590602561026934038191389168, −22.032969062921444167054656441927, −21.84678370556842052914314180448, −20.54065330764430545999171248054, −19.63690647282576431761926145034, −18.71711424196461945897344913137, −17.93434033456423639769481467759, −16.96629224797543154854565298920, −16.324573511293061263481980223908, −15.46130033356068742412266682098, −14.64542386740984051319901265792, −13.88483467734915038974522144031, −12.84431574194201559054099146656, −11.975121022020758935051129069772, −11.431484430741817455023095662092, −11.11745897863082648381152678624, −9.132354998097660856443188454551, −8.14750747478262223590470324663, −7.346733535538171497387565123821, −6.5771584443286868450776461422, −5.81842466546233100289810391167, −4.57850130658969263110560411672, −4.2060768415727685504107192075, −2.658845796577174387871690885978, −1.58809015819819318382354596390,
0.57378565472554356592698962226, 1.77264758985675419877276878815, 3.58947721062004575720448366418, 4.04859240120007690246247170599, 4.71455209599575216554736574583, 5.81240687332993290601194466282, 6.63640838485986544571542980609, 7.77179441341731965560865139662, 8.90796704686102301866181714910, 10.28977063892506051638088821514, 10.776910765065577749595456173080, 11.4937791866851250065291796409, 12.26510126175705478884706371376, 12.92150308816001190830120111024, 14.277905609147200441930876956923, 14.93882982564556540008173446144, 15.55832046167248347525969917284, 16.532111294740419967201131031368, 17.23829980436732928993537999976, 18.26777422359987967369664471437, 19.47806631909210098538330090882, 20.093997936212383206543767876801, 20.76580104823527934679131039015, 21.601626352478810238948621906815, 22.47390486151207028508302825214