L(s) = 1 | + (0.981 + 0.190i)2-s + (−0.435 − 0.900i)3-s + (0.927 + 0.373i)4-s + (0.149 − 0.988i)5-s + (−0.256 − 0.966i)6-s + (0.662 − 0.749i)7-s + (0.839 + 0.542i)8-s + (−0.620 + 0.784i)9-s + (0.334 − 0.942i)10-s + (−0.0682 − 0.997i)12-s + (0.385 − 0.922i)13-s + (0.792 − 0.609i)14-s + (−0.955 + 0.295i)15-s + (0.721 + 0.692i)16-s + (0.740 − 0.672i)17-s + (−0.758 + 0.652i)18-s + ⋯ |
L(s) = 1 | + (0.981 + 0.190i)2-s + (−0.435 − 0.900i)3-s + (0.927 + 0.373i)4-s + (0.149 − 0.988i)5-s + (−0.256 − 0.966i)6-s + (0.662 − 0.749i)7-s + (0.839 + 0.542i)8-s + (−0.620 + 0.784i)9-s + (0.334 − 0.942i)10-s + (−0.0682 − 0.997i)12-s + (0.385 − 0.922i)13-s + (0.792 − 0.609i)14-s + (−0.955 + 0.295i)15-s + (0.721 + 0.692i)16-s + (0.740 − 0.672i)17-s + (−0.758 + 0.652i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.759842128 - 3.306196403i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.759842128 - 3.306196403i\) |
\(L(1)\) |
\(\approx\) |
\(1.684197433 - 0.9416677967i\) |
\(L(1)\) |
\(\approx\) |
\(1.684197433 - 0.9416677967i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 47 | \( 1 \) |
good | 2 | \( 1 + (0.981 + 0.190i)T \) |
| 3 | \( 1 + (-0.435 - 0.900i)T \) |
| 5 | \( 1 + (0.149 - 0.988i)T \) |
| 7 | \( 1 + (0.662 - 0.749i)T \) |
| 13 | \( 1 + (0.385 - 0.922i)T \) |
| 17 | \( 1 + (0.740 - 0.672i)T \) |
| 19 | \( 1 + (-0.986 - 0.163i)T \) |
| 23 | \( 1 + (0.682 - 0.730i)T \) |
| 29 | \( 1 + (-0.758 + 0.652i)T \) |
| 31 | \( 1 + (0.881 - 0.472i)T \) |
| 37 | \( 1 + (0.792 + 0.609i)T \) |
| 41 | \( 1 + (-0.360 + 0.932i)T \) |
| 43 | \( 1 + (0.0682 - 0.997i)T \) |
| 53 | \( 1 + (0.998 + 0.0546i)T \) |
| 59 | \( 1 + (-0.969 - 0.243i)T \) |
| 61 | \( 1 + (0.282 - 0.959i)T \) |
| 67 | \( 1 + (-0.917 + 0.398i)T \) |
| 71 | \( 1 + (-0.981 + 0.190i)T \) |
| 73 | \( 1 + (-0.554 + 0.832i)T \) |
| 79 | \( 1 + (-0.946 - 0.321i)T \) |
| 83 | \( 1 + (-0.410 + 0.911i)T \) |
| 89 | \( 1 + (0.460 + 0.887i)T \) |
| 97 | \( 1 + (0.721 - 0.692i)T \) |
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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
−23.27980081394826115680725037513, −22.82071247515638327088759471594, −21.69084174309132179557178844262, −21.41329790358866199093364407420, −20.86862165460776327159950831010, −19.41804616063481793087583621161, −18.756732959244603079472669647748, −17.57243795527938435754545204280, −16.67026520921641555588634914002, −15.6440648047031443704699224630, −14.87261134323799474311827072753, −14.56175980057445190936519329031, −13.45629246410532276144996296059, −12.1456190247385101297493030086, −11.47603796900518629728976746232, −10.826336153448059713603442033646, −10.02021158916517003639194357844, −8.86910557852675546404447955588, −7.46156345933502639792096188866, −6.19550402643351683354049591170, −5.79281065279160038175866231921, −4.6045086093792871764048314752, −3.77676620038919028014197545442, −2.76168997164120734787872561443, −1.643441598902251128390431669143,
0.69810328301306221514135502319, 1.580070076090827355877078907434, 2.8519266249117480542695352096, 4.314344817434231030843229509833, 5.09906130731261837250620035322, 5.87920598806740769075286101313, 6.94461140987681917719419158759, 7.85653512992646034035990549263, 8.502127195607057528556157943218, 10.30776179841975860799159925905, 11.20346902464556764313498412600, 12.01739116136888637842298575331, 12.93446047070775835181525672000, 13.33194494796360152913906751530, 14.24755981235223888499808208818, 15.217924565583128952680394039543, 16.52044178060830815152774168440, 16.899527952517466428775367675862, 17.71001780150774330294898367122, 18.86088225091764445087582491830, 20.10878607107288280535391127087, 20.47904357379729130067484296495, 21.36174437649456118136183577798, 22.48268672836354683548187805047, 23.38323854034836685238552620475