| L(s) = 1 | − i·3-s − 4i·7-s + 2·9-s − 6·11-s + i·13-s − 2·19-s − 4·21-s − i·23-s − 5i·27-s − 9·29-s + 5·31-s + 6i·33-s + 2i·37-s + 39-s − 9·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.51i·7-s + 0.666·9-s − 1.80·11-s + 0.277i·13-s − 0.458·19-s − 0.872·21-s − 0.208i·23-s − 0.962i·27-s − 1.67·29-s + 0.898·31-s + 1.04i·33-s + 0.328i·37-s + 0.160·39-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5006844626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5006844626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.176555094799177581718049673505, −7.79196115165720721342594144563, −7.08481640846764953556820124622, −6.49724703103641088089339742306, −5.32333015488759902850864674777, −4.51362422315349376901523685725, −3.70542405516851421884586547257, −2.51207310967288742439514606337, −1.42286366840267562640283688579, −0.16339321070087693387632007168,
1.93780567880003502341009651791, 2.73894347692831500873402811056, 3.72345083160389031744303056907, 4.94813984113802290231143898505, 5.31770153608145184262237174947, 6.13242561854123591160145229221, 7.25909361378451608186088683608, 8.019193622759623204402769801509, 8.716885702611286570852312680671, 9.460606345505189938854885297554
