| L(s) = 1 | − 3-s + (−1.22 + 1.87i)5-s − 2·9-s − 4.89·11-s + 4.58i·13-s + (1.22 − 1.87i)15-s + 2.44·17-s − 2.44·19-s + (3 − 3.74i)23-s + (−2 − 4.58i)25-s + 5·27-s − 3·29-s + 4.58i·31-s + 4.89·33-s − 4.89·37-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (−0.547 + 0.836i)5-s − 0.666·9-s − 1.47·11-s + 1.27i·13-s + (0.316 − 0.483i)15-s + 0.594·17-s − 0.561·19-s + (0.625 − 0.780i)23-s + (−0.400 − 0.916i)25-s + 0.962·27-s − 0.557·29-s + 0.823i·31-s + 0.852·33-s − 0.805·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4042211149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4042211149\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.22 - 1.87i)T \) |
| 23 | \( 1 + (-3 + 3.74i)T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.58iT - 13T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 4.58iT - 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 9T + 47T^{2} \) |
| 53 | \( 1 - 4.89T + 53T^{2} \) |
| 59 | \( 1 + 9.16iT - 59T^{2} \) |
| 61 | \( 1 - 11.2iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 13.7iT - 71T^{2} \) |
| 73 | \( 1 - 4.58iT - 73T^{2} \) |
| 79 | \( 1 + 7.34T + 79T^{2} \) |
| 83 | \( 1 - 3.74iT - 83T^{2} \) |
| 89 | \( 1 + 3.74iT - 89T^{2} \) |
| 97 | \( 1 + 9.79T + 97T^{2} \) |
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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
−8.905879925639160074308184821674, −8.367985122221895970449467410793, −7.29693436948795381833854019168, −6.85178819604670818816212913684, −5.84045281471823175332670279333, −5.13834659140349665586228058570, −4.12337135156646103650503542461, −3.06603672275557544851604930485, −2.20348192782250517393963725080, −0.20962875270786755658855976663,
0.875705896901860638765566280299, 2.58088685929816378958897012716, 3.51864831528984031654105122518, 4.72681736403762576422305016699, 5.51067007451890488921641209216, 5.76134302265987595293820498578, 7.24481117532920223571925179785, 7.927617361761667570248309964617, 8.439510214768079925730990436696, 9.358221783978074074212656513828
