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
−25.4484503749328110177196680704, −24.7361570627664272543722965199, −23.60882728495903381678732303666, −22.1290875763838916947310074270, −21.43232488061507508471861407482, −20.84305987053159840438753988109, −19.82996405693112664225647880493, −18.79200739657776014029027422174, −17.87264379256810551766963151803, −16.796687538961063476183743135, −16.468734173286861425683319801259, −15.338760643275508828235800122310, −14.32676387510602036910052960202, −13.3368005615931728935872465840, −11.544239432617936544025468262181, −11.065793045804493239337152748887, −9.90223332086882740948076544757, −9.23880321926159444930431652827, −8.58637380499668026935134482886, −6.99490065485122487416323227854, −5.982930145617967908734131472446, −4.93125832791676840034835973289, −3.24780120474281393298424073653, −2.235592688447811375653258821792, −0.57188728472683410728204360526,
1.169805011957107214398396836289, 1.95106079803916044176297306184, 3.08512626982350952776889053748, 5.41591548530614723161241924222, 6.28072609462002170018562686201, 7.24273814014498593010515727608, 8.155750723401929983138159988255, 9.18376142791688544411802152163, 10.205836369858183463712413906101, 11.05055935713582216903740478404, 12.56426690691076525576967934898, 12.824967845853139949568798368986, 14.39257421434502557604803530932, 15.12401306906261956294001228270, 16.72268837795427718367580297149, 17.30356185938159205088407469707, 18.10899808043816626964410450192, 18.65715968772054974462181897042, 19.90131873850427421470742042557, 20.42827234458333910187694841794, 21.55707631452237126092065153854, 22.79627114455007876860879770137, 23.80249031190228844972799845607, 24.876551787690887051548309322644, 25.38010683091884008707982712740