| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 1.70i·7-s + i·8-s − 9-s − 1.70·11-s − i·12-s − i·13-s + 1.70·14-s + 16-s − 5.70i·17-s + i·18-s − 4.70·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.643i·7-s + 0.353i·8-s − 0.333·9-s − 0.513·11-s − 0.288i·12-s − 0.277i·13-s + 0.454·14-s + 0.250·16-s − 1.38i·17-s + 0.235i·18-s − 1.07·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9087820214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9087820214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 1.70iT - 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 17 | \( 1 + 5.70iT - 17T^{2} \) |
| 19 | \( 1 + 4.70T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6.40T + 29T^{2} \) |
| 31 | \( 1 - 9.10T + 31T^{2} \) |
| 37 | \( 1 + 4.70iT - 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 - 1.40iT - 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 + 10.4iT - 53T^{2} \) |
| 59 | \( 1 - 3.70T + 59T^{2} \) |
| 61 | \( 1 + 5.10T + 61T^{2} \) |
| 67 | \( 1 + 6.40iT - 67T^{2} \) |
| 71 | \( 1 - 4.70T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 + 0.701T + 79T^{2} \) |
| 83 | \( 1 + 4.29iT - 83T^{2} \) |
| 89 | \( 1 - 1.40T + 89T^{2} \) |
| 97 | \( 1 - 15.4iT - 97T^{2} \) |
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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
−9.098344133612582712863648643365, −8.398824621956315270061346840663, −7.56314425882558681482669381367, −6.41065079232029427755810455797, −5.43817931437966307561421473870, −4.84429867792614208156578714816, −3.87866923938208834774460505366, −2.86436314574545733966067158495, −2.14786577975201195181248835699, −0.34777557407555066945862384084,
1.23035842636700535161064277556, 2.52309446317367255329252021286, 3.86648987870661169958224495599, 4.54865814961900231855720767440, 5.72037487966762957851988895343, 6.33749469287660537007995285816, 7.06718613940238667045429929336, 7.892981010869634570080017333537, 8.363241572463843724273222181126, 9.229156660401381971842556786942
