| L(s) = 1 | + i·2-s − 3.44i·3-s − 4-s + 3.44·6-s − 2.44i·7-s − i·8-s − 8.89·9-s − 1.44·11-s + 3.44i·12-s − 4.44i·13-s + 2.44·14-s + 16-s − 1.44i·17-s − 8.89i·18-s + 5·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 1.99i·3-s − 0.5·4-s + 1.40·6-s − 0.925i·7-s − 0.353i·8-s − 2.96·9-s − 0.437·11-s + 0.995i·12-s − 1.23i·13-s + 0.654·14-s + 0.250·16-s − 0.351i·17-s − 2.09i·18-s + 1.14·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7819261305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7819261305\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
| good | 3 | \( 1 + 3.44iT - 3T^{2} \) |
| 7 | \( 1 + 2.44iT - 7T^{2} \) |
| 11 | \( 1 + 1.44T + 11T^{2} \) |
| 13 | \( 1 + 4.44iT - 13T^{2} \) |
| 17 | \( 1 + 1.44iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + 8.89T + 29T^{2} \) |
| 31 | \( 1 + 0.449T + 31T^{2} \) |
| 41 | \( 1 - T + 41T^{2} \) |
| 43 | \( 1 - 10.8iT - 43T^{2} \) |
| 47 | \( 1 + 9.79iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 + 1.55T + 61T^{2} \) |
| 67 | \( 1 - 9.44iT - 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 6.79iT - 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 1.44iT - 83T^{2} \) |
| 89 | \( 1 - 0.348T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.318580865048947843111042242965, −7.70559013863931464521431841479, −7.35499684294625359338516012553, −6.65339798989996639850950605168, −5.71236086579124226334486085251, −5.20870283699706046340956767328, −3.56930591230658932043545755396, −2.61013188441209126286303639610, −1.22848237137286361004802327564, −0.30544122440956292475402584651,
2.10957037291997727923498760605, 3.11656266968903612674162587902, 3.83810086970348070372001989742, 4.66832141511726138909786244960, 5.42206936754834365309303979514, 5.98734591435811200403411599285, 7.61764887035837150882021165278, 8.668064125974959772806976235794, 9.320660675724595626555490523286, 9.463401726632281992266270075166
