L(s) = 1 | + (0.213 − 0.976i)2-s + (0.932 + 0.361i)3-s + (−0.908 − 0.417i)4-s + (−0.0307 − 0.999i)5-s + (0.552 − 0.833i)6-s + (0.992 − 0.122i)7-s + (−0.602 + 0.798i)8-s + (0.739 + 0.673i)9-s + (−0.982 − 0.183i)10-s + (−0.952 + 0.303i)11-s + (−0.696 − 0.717i)12-s + (−0.602 − 0.798i)13-s + (0.0922 − 0.995i)14-s + (0.332 − 0.943i)15-s + (0.650 + 0.759i)16-s + (0.552 + 0.833i)17-s + ⋯ |
L(s) = 1 | + (0.213 − 0.976i)2-s + (0.932 + 0.361i)3-s + (−0.908 − 0.417i)4-s + (−0.0307 − 0.999i)5-s + (0.552 − 0.833i)6-s + (0.992 − 0.122i)7-s + (−0.602 + 0.798i)8-s + (0.739 + 0.673i)9-s + (−0.982 − 0.183i)10-s + (−0.952 + 0.303i)11-s + (−0.696 − 0.717i)12-s + (−0.602 − 0.798i)13-s + (0.0922 − 0.995i)14-s + (0.332 − 0.943i)15-s + (0.650 + 0.759i)16-s + (0.552 + 0.833i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0861 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0861 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.025307017 - 0.9404909195i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.025307017 - 0.9404909195i\) |
\(L(1)\) |
\(\approx\) |
\(1.183324553 - 0.7002281375i\) |
\(L(1)\) |
\(\approx\) |
\(1.183324553 - 0.7002281375i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 103 | \( 1 \) |
good | 2 | \( 1 + (0.213 - 0.976i)T \) |
| 3 | \( 1 + (0.932 + 0.361i)T \) |
| 5 | \( 1 + (-0.0307 - 0.999i)T \) |
| 7 | \( 1 + (0.992 - 0.122i)T \) |
| 11 | \( 1 + (-0.952 + 0.303i)T \) |
| 13 | \( 1 + (-0.602 - 0.798i)T \) |
| 17 | \( 1 + (0.552 + 0.833i)T \) |
| 19 | \( 1 + (-0.779 - 0.626i)T \) |
| 23 | \( 1 + (0.739 - 0.673i)T \) |
| 29 | \( 1 + (0.881 + 0.473i)T \) |
| 31 | \( 1 + (-0.982 + 0.183i)T \) |
| 37 | \( 1 + (-0.273 + 0.961i)T \) |
| 41 | \( 1 + (-0.0307 + 0.999i)T \) |
| 43 | \( 1 + (-0.696 + 0.717i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.153 + 0.988i)T \) |
| 59 | \( 1 + (0.992 + 0.122i)T \) |
| 61 | \( 1 + (0.445 - 0.895i)T \) |
| 67 | \( 1 + (-0.389 - 0.920i)T \) |
| 71 | \( 1 + (0.881 - 0.473i)T \) |
| 73 | \( 1 + (-0.850 - 0.526i)T \) |
| 79 | \( 1 + (-0.850 + 0.526i)T \) |
| 83 | \( 1 + (-0.389 + 0.920i)T \) |
| 89 | \( 1 + (0.0922 - 0.995i)T \) |
| 97 | \( 1 + (0.552 - 0.833i)T \) |
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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
−30.399634688133850923043968793818, −29.36315645281229252657043634050, −27.29767424468082267945283047799, −26.81392260964469163572458775598, −25.80026411176304547228177930841, −25.00354814320790003977976420888, −23.91540080149705827710362559692, −23.195743985292419241493708442496, −21.620240101578327711318038294296, −20.99441042749617539441315214914, −19.10271461750418872314940396795, −18.48708873894265037202092340815, −17.50128041111661088365931340700, −15.923978726361334020274259077603, −14.789741580731644836442725400, −14.32637161054451524654063777277, −13.31039662320661956885180141026, −11.834423130578864713010937399193, −10.1218538440763744985910435159, −8.734785510364582196052141461066, −7.66801407567204672014924405311, −6.9653762196433403491586786880, −5.335191321461767873690410742123, −3.78555174668177608568849217555, −2.354058953019557096974311067732,
1.55642907694174682032742180160, 2.89391081364508771844549457207, 4.51621594234409596265161903629, 5.10769188537599999681886957401, 7.963072322244694651281885897495, 8.61918613562717689156398500884, 9.94707380147559308088128608195, 10.8739663757077746128786051776, 12.54248506064943789148597206877, 13.14826342552255915747580242998, 14.49464701705331714171443392710, 15.30573725344892673806277178558, 17.001946693658152394812554589002, 18.17804979567639220289488706706, 19.509766383521047621718252475329, 20.29763704481455178955547574146, 21.04670811605371138460349663953, 21.7114505785388442143403025912, 23.42603492275696976347466530728, 24.2448895316103554271175532492, 25.47780328305789779617479878124, 26.836767546783987511341775934265, 27.6150237166539265027458198806, 28.396869622734826920364850200626, 29.7216505592317360329111572260