L(s) = 1 | − 2-s + i·3-s + 4-s + 5-s − i·6-s − 8-s − 9-s − 10-s − i·11-s + i·12-s + i·13-s + i·15-s + 16-s − i·17-s + 18-s − i·19-s + ⋯ |
L(s) = 1 | − 2-s + i·3-s + 4-s + 5-s − i·6-s − 8-s − 9-s − 10-s − i·11-s + i·12-s + i·13-s + i·15-s + 16-s − i·17-s + 18-s − i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.244374321 - 0.2119214532i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.244374321 - 0.2119214532i\) |
\(L(1)\) |
\(\approx\) |
\(0.8227200263 + 0.1044464862i\) |
\(L(1)\) |
\(\approx\) |
\(0.8227200263 + 0.1044464862i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - iT \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - iT \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - iT \) |
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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
−25.38010683091884008707982712740, −24.876551787690887051548309322644, −23.80249031190228844972799845607, −22.79627114455007876860879770137, −21.55707631452237126092065153854, −20.42827234458333910187694841794, −19.90131873850427421470742042557, −18.65715968772054974462181897042, −18.10899808043816626964410450192, −17.30356185938159205088407469707, −16.72268837795427718367580297149, −15.12401306906261956294001228270, −14.39257421434502557604803530932, −12.824967845853139949568798368986, −12.56426690691076525576967934898, −11.05055935713582216903740478404, −10.205836369858183463712413906101, −9.18376142791688544411802152163, −8.155750723401929983138159988255, −7.24273814014498593010515727608, −6.28072609462002170018562686201, −5.41591548530614723161241924222, −3.08512626982350952776889053748, −1.95106079803916044176297306184, −1.169805011957107214398396836289,
0.57188728472683410728204360526, 2.235592688447811375653258821792, 3.24780120474281393298424073653, 4.93125832791676840034835973289, 5.982930145617967908734131472446, 6.99490065485122487416323227854, 8.58637380499668026935134482886, 9.23880321926159444930431652827, 9.90223332086882740948076544757, 11.065793045804493239337152748887, 11.544239432617936544025468262181, 13.3368005615931728935872465840, 14.32676387510602036910052960202, 15.338760643275508828235800122310, 16.468734173286861425683319801259, 16.796687538961063476183743135, 17.87264379256810551766963151803, 18.79200739657776014029027422174, 19.82996405693112664225647880493, 20.84305987053159840438753988109, 21.43232488061507508471861407482, 22.1290875763838916947310074270, 23.60882728495903381678732303666, 24.7361570627664272543722965199, 25.4484503749328110177196680704