L(s) = 1 | + (0.391 − 0.920i)2-s + (−0.872 − 0.489i)3-s + (−0.694 − 0.719i)4-s + (−0.457 − 0.889i)5-s + (−0.791 + 0.611i)6-s + (0.520 + 0.853i)7-s + (−0.934 + 0.357i)8-s + (0.520 + 0.853i)9-s + (−0.997 + 0.0729i)10-s + (0.957 + 0.288i)11-s + (0.252 + 0.967i)12-s + (0.391 + 0.920i)13-s + (0.989 − 0.145i)14-s + (−0.0365 + 0.999i)15-s + (−0.0365 + 0.999i)16-s + (0.957 + 0.288i)17-s + ⋯ |
L(s) = 1 | + (0.391 − 0.920i)2-s + (−0.872 − 0.489i)3-s + (−0.694 − 0.719i)4-s + (−0.457 − 0.889i)5-s + (−0.791 + 0.611i)6-s + (0.520 + 0.853i)7-s + (−0.934 + 0.357i)8-s + (0.520 + 0.853i)9-s + (−0.997 + 0.0729i)10-s + (0.957 + 0.288i)11-s + (0.252 + 0.967i)12-s + (0.391 + 0.920i)13-s + (0.989 − 0.145i)14-s + (−0.0365 + 0.999i)15-s + (−0.0365 + 0.999i)16-s + (0.957 + 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9662338237 - 0.4320936949i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9662338237 - 0.4320936949i\) |
\(L(1)\) |
\(\approx\) |
\(0.8244965691 - 0.4543034262i\) |
\(L(1)\) |
\(\approx\) |
\(0.8244965691 - 0.4543034262i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 431 | \( 1 \) |
good | 2 | \( 1 + (0.391 - 0.920i)T \) |
| 3 | \( 1 + (-0.872 - 0.489i)T \) |
| 5 | \( 1 + (-0.457 - 0.889i)T \) |
| 7 | \( 1 + (0.520 + 0.853i)T \) |
| 11 | \( 1 + (0.957 + 0.288i)T \) |
| 13 | \( 1 + (0.391 + 0.920i)T \) |
| 17 | \( 1 + (0.957 + 0.288i)T \) |
| 19 | \( 1 + (-0.457 + 0.889i)T \) |
| 23 | \( 1 + (0.905 + 0.424i)T \) |
| 29 | \( 1 + (-0.694 + 0.719i)T \) |
| 31 | \( 1 + (-0.791 - 0.611i)T \) |
| 37 | \( 1 + (0.989 - 0.145i)T \) |
| 41 | \( 1 + (-0.0365 + 0.999i)T \) |
| 43 | \( 1 + (-0.997 - 0.0729i)T \) |
| 47 | \( 1 + (-0.181 + 0.983i)T \) |
| 53 | \( 1 + (-0.181 - 0.983i)T \) |
| 59 | \( 1 + (0.391 - 0.920i)T \) |
| 61 | \( 1 + (-0.976 + 0.217i)T \) |
| 67 | \( 1 + (0.989 - 0.145i)T \) |
| 71 | \( 1 + (-0.181 + 0.983i)T \) |
| 73 | \( 1 + (0.109 - 0.994i)T \) |
| 79 | \( 1 + (0.744 + 0.667i)T \) |
| 83 | \( 1 + (-0.872 - 0.489i)T \) |
| 89 | \( 1 + (-0.322 + 0.946i)T \) |
| 97 | \( 1 + (0.744 - 0.667i)T \) |
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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
−23.97487370951278081874138453541, −23.21858043716914911455807749582, −22.81934413226768479564127472309, −21.97140446559804446489454856977, −21.19122780951890907248807797453, −20.06988539656985376257322364252, −18.69674299865406242639200582033, −17.93344595385399785787203496001, −17.04809351645063827047653481370, −16.55938034882662961317357648850, −15.36731948218567830481664299982, −14.87020171697986311093773253150, −13.97942419261092744371485541182, −12.873952280974710083134195353000, −11.72552752824830606476976431536, −11.01888987644141686087983773622, −10.09057110096404084601317082828, −8.83345687201799350539485681822, −7.59263579563306685666111141792, −6.90171042690845747997485478697, −6.02202259453369246655452808653, −4.96766350651138172225207399244, −3.97488366739698638452295529, −3.27390504668661072967491049774, −0.71227535084211141423132652443,
1.30561560302372750990053885508, 1.80403038370584194675872445519, 3.668544605867828385499355305909, 4.638870879007723975515490844986, 5.45380581638382815367033160690, 6.33682041611659296328503411558, 7.85762976869597416916032191117, 8.907303007314542251385014638647, 9.74529841765283770734742329960, 11.33482806677574456499611015775, 11.48983387141587631525925436272, 12.50856734295016840110753292417, 12.90211867534665493171400855353, 14.27046525842821102866197998602, 15.06900377669173668442076022281, 16.45787065062549188333460819825, 17.06243606617450696996885307708, 18.252823809468315561877505384125, 18.891478182615746925299948399987, 19.61657950344085793930031153913, 20.77130731612483236403761235386, 21.438648878931523104270456381600, 22.22467750163696238573004941059, 23.285754619107209242271964054788, 23.70352619753832629291601872327