L(s) = 1 | + (−0.697 − 0.716i)2-s + (0.658 − 0.752i)3-s + (−0.0266 + 0.999i)4-s + (0.910 − 0.413i)5-s + (−0.998 + 0.0532i)6-s + (−0.769 − 0.638i)7-s + (0.734 − 0.678i)8-s + (−0.132 − 0.991i)9-s + (−0.931 − 0.364i)10-s + (0.484 + 0.874i)11-s + (0.734 + 0.678i)12-s + (−0.769 + 0.638i)13-s + (0.0797 + 0.996i)14-s + (0.288 − 0.957i)15-s + (−0.998 − 0.0532i)16-s + (0.484 − 0.874i)17-s + ⋯ |
L(s) = 1 | + (−0.697 − 0.716i)2-s + (0.658 − 0.752i)3-s + (−0.0266 + 0.999i)4-s + (0.910 − 0.413i)5-s + (−0.998 + 0.0532i)6-s + (−0.769 − 0.638i)7-s + (0.734 − 0.678i)8-s + (−0.132 − 0.991i)9-s + (−0.931 − 0.364i)10-s + (0.484 + 0.874i)11-s + (0.734 + 0.678i)12-s + (−0.769 + 0.638i)13-s + (0.0797 + 0.996i)14-s + (0.288 − 0.957i)15-s + (−0.998 − 0.0532i)16-s + (0.484 − 0.874i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1970584688 - 1.166612179i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1970584688 - 1.166612179i\) |
\(L(1)\) |
\(\approx\) |
\(0.7016454628 - 0.6685728018i\) |
\(L(1)\) |
\(\approx\) |
\(0.7016454628 - 0.6685728018i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 709 | \( 1 \) |
good | 2 | \( 1 + (-0.697 - 0.716i)T \) |
| 3 | \( 1 + (0.658 - 0.752i)T \) |
| 5 | \( 1 + (0.910 - 0.413i)T \) |
| 7 | \( 1 + (-0.769 - 0.638i)T \) |
| 11 | \( 1 + (0.484 + 0.874i)T \) |
| 13 | \( 1 + (-0.769 + 0.638i)T \) |
| 17 | \( 1 + (0.484 - 0.874i)T \) |
| 19 | \( 1 + (0.734 - 0.678i)T \) |
| 23 | \( 1 + (-0.964 + 0.263i)T \) |
| 29 | \( 1 + (-0.617 - 0.786i)T \) |
| 31 | \( 1 + (0.288 - 0.957i)T \) |
| 37 | \( 1 + (-0.769 - 0.638i)T \) |
| 41 | \( 1 + (-0.237 + 0.971i)T \) |
| 43 | \( 1 + (0.977 - 0.211i)T \) |
| 47 | \( 1 + (-0.964 + 0.263i)T \) |
| 53 | \( 1 + (0.288 - 0.957i)T \) |
| 59 | \( 1 + (0.734 - 0.678i)T \) |
| 61 | \( 1 + (-0.530 + 0.847i)T \) |
| 67 | \( 1 + (-0.339 - 0.940i)T \) |
| 71 | \( 1 + (-0.931 + 0.364i)T \) |
| 73 | \( 1 + (0.861 - 0.507i)T \) |
| 79 | \( 1 + (0.802 - 0.596i)T \) |
| 83 | \( 1 + (0.185 - 0.982i)T \) |
| 89 | \( 1 + (0.994 - 0.106i)T \) |
| 97 | \( 1 + (0.388 - 0.921i)T \) |
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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
−22.65414447475075649827659752758, −22.224089425512545510425731175916, −21.473774149194962380025252712691, −20.39940848715206520290543625049, −19.50186619842963845343590729838, −18.952637893753871284976379997693, −18.092452679174194047752653318511, −17.07609157149291656572740454093, −16.418554175775195551170262492327, −15.679402524514445976751507673107, −14.74627801253810346565796697934, −14.2587069620178246154257953454, −13.451902138613889274430022550198, −12.186180909968823952179110936253, −10.67770953546633737956957377530, −10.17680542567189461073153908223, −9.465188875411103838660905407705, −8.75599014483623876743308583334, −7.90779871359960508469066632315, −6.74066541191204936996111242513, −5.75255589953056349753474588064, −5.30584311556104932132670734928, −3.63669524449607355666824401602, −2.72189490448023450669277623049, −1.59474461536049726389637309509,
0.644698996086902090372528889617, 1.79412217760186146746848782224, 2.47757443435077327920936579736, 3.57034286419288725731220107818, 4.65520596160843743795779871986, 6.27621619526258561117154820939, 7.170548652804748690221576168351, 7.731215534726349619560566141595, 9.13160732125192903024412904968, 9.565746580744071579151265179398, 10.02880921123819866213934453057, 11.6366219608141990440891904371, 12.23547885971746897098495040838, 13.14980927214868365224607032498, 13.66547662146234385012064068474, 14.50663194135924298123323350046, 15.968694959984178240911243403722, 16.85842074817165961079148676467, 17.55665203743623670119841968268, 18.167051720729691441630975570787, 19.19669792772193702143133937961, 19.76975519310292234861570684742, 20.46132116011628538517412969839, 21.034135097362413680845180674468, 22.20169862460917265685441978471