| L(s) = 1 | + i·5-s − 1.04i·7-s − 1.61i·11-s + (3.25 − 1.54i)13-s − 1.04·17-s + 5.52i·19-s + 4.13·23-s − 25-s − 4.53·29-s + 3.08i·31-s + 1.04·35-s + 9.65i·37-s − 0.380i·41-s + 5.89·49-s + 3.48·53-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 0.396i·7-s − 0.488i·11-s + (0.903 − 0.427i)13-s − 0.254·17-s + 1.26i·19-s + 0.862·23-s − 0.200·25-s − 0.842·29-s + 0.554i·31-s + 0.177·35-s + 1.58i·37-s − 0.0593i·41-s + 0.842·49-s + 0.478·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.933025721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.933025721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-3.25 + 1.54i)T \) |
| good | 7 | \( 1 + 1.04iT - 7T^{2} \) |
| 11 | \( 1 + 1.61iT - 11T^{2} \) |
| 17 | \( 1 + 1.04T + 17T^{2} \) |
| 19 | \( 1 - 5.52iT - 19T^{2} \) |
| 23 | \( 1 - 4.13T + 23T^{2} \) |
| 29 | \( 1 + 4.53T + 29T^{2} \) |
| 31 | \( 1 - 3.08iT - 31T^{2} \) |
| 37 | \( 1 - 9.65iT - 37T^{2} \) |
| 41 | \( 1 + 0.380iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 3.48T + 53T^{2} \) |
| 59 | \( 1 + 5.27iT - 59T^{2} \) |
| 61 | \( 1 - 4.89T + 61T^{2} \) |
| 67 | \( 1 + 10.0iT - 67T^{2} \) |
| 71 | \( 1 + 6.13iT - 71T^{2} \) |
| 73 | \( 1 - 0.650iT - 73T^{2} \) |
| 79 | \( 1 + 1.10T + 79T^{2} \) |
| 83 | \( 1 + 7.23iT - 83T^{2} \) |
| 89 | \( 1 - 7.61iT - 89T^{2} \) |
| 97 | \( 1 - 3.02iT - 97T^{2} \) |
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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
−8.323859489784058561590520648302, −7.66845419545136045191806787032, −6.87969891123155408830282428301, −6.18720370315730971388345509763, −5.55233455408881969299552870197, −4.61382356257265683429239928200, −3.58388025491272542736489743952, −3.22317361393370902945739283980, −1.93563451250914493319480268424, −0.889442030354138803687032952125,
0.69430113462722915202262282105, 1.88390510184567771292475177978, 2.71384684849235405628072767659, 3.85792748704375173452436193557, 4.47836694840910231732315672751, 5.37960627299987627062905726050, 5.94968801296260417997569645234, 6.96662225702371926848023827476, 7.36726645622785822716704168921, 8.477278415413800262554007937288
