| L(s) = 1 | − 1.44·3-s + i·5-s + (2.48 + 0.919i)7-s − 0.912·9-s + 5.66i·11-s + 5.14i·13-s − 1.44i·15-s + 1.65i·17-s + 1.33·19-s + (−3.58 − 1.32i)21-s − 3.62i·23-s − 25-s + 5.65·27-s + 0.0482·29-s − 3.08·31-s + ⋯ |
| L(s) = 1 | − 0.834·3-s + 0.447i·5-s + (0.937 + 0.347i)7-s − 0.304·9-s + 1.70i·11-s + 1.42i·13-s − 0.373i·15-s + 0.401i·17-s + 0.307·19-s + (−0.782 − 0.289i)21-s − 0.755i·23-s − 0.200·25-s + 1.08·27-s + 0.00895·29-s − 0.554·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8757093330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8757093330\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-2.48 - 0.919i)T \) |
| good | 3 | \( 1 + 1.44T + 3T^{2} \) |
| 11 | \( 1 - 5.66iT - 11T^{2} \) |
| 13 | \( 1 - 5.14iT - 13T^{2} \) |
| 17 | \( 1 - 1.65iT - 17T^{2} \) |
| 19 | \( 1 - 1.33T + 19T^{2} \) |
| 23 | \( 1 + 3.62iT - 23T^{2} \) |
| 29 | \( 1 - 0.0482T + 29T^{2} \) |
| 31 | \( 1 + 3.08T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 2.37iT - 41T^{2} \) |
| 43 | \( 1 + 6.18iT - 43T^{2} \) |
| 47 | \( 1 - 7.75T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 5.32T + 59T^{2} \) |
| 61 | \( 1 + 2.30iT - 61T^{2} \) |
| 67 | \( 1 - 0.207iT - 67T^{2} \) |
| 71 | \( 1 + 6.67iT - 71T^{2} \) |
| 73 | \( 1 - 3.42iT - 73T^{2} \) |
| 79 | \( 1 + 1.73iT - 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 4.28iT - 89T^{2} \) |
| 97 | \( 1 - 12.1iT - 97T^{2} \) |
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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
−9.323654142628362522493596335078, −8.699473187397855328873078062340, −7.68384094236676893498579426575, −6.93428738367258217720814098795, −6.36424899704288570341670763137, −5.28495838243223808464607563094, −4.76485508912444883520788856465, −3.88893899398177097178181004113, −2.36692396046649256745355333678, −1.65050537221761179282651522240,
0.36194111110316736206149337778, 1.26201660739763057726152459983, 2.90612733415007927667511084551, 3.75713750403757963135499044330, 5.09680225885385087697568998861, 5.40724128789146535755239585469, 6.05240191134522596184701296245, 7.19377225677031939498177534431, 8.084596509741864397626203773236, 8.494094309705624130750492561804
