| L(s) = 1 | + i·2-s − 4-s + (−1.41 − 2.23i)7-s − i·8-s + 5.65i·11-s − 4.47i·13-s + (2.23 − 1.41i)14-s + 16-s − 3.16·17-s − 3.16i·19-s − 5.65·22-s + 4i·23-s + 4.47·26-s + (1.41 + 2.23i)28-s − 2.82i·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.534 − 0.845i)7-s − 0.353i·8-s + 1.70i·11-s − 1.24i·13-s + (0.597 − 0.377i)14-s + 0.250·16-s − 0.766·17-s − 0.725i·19-s − 1.20·22-s + 0.834i·23-s + 0.877·26-s + (0.267 + 0.422i)28-s − 0.525i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0515 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0515 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.290090439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.290090439\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.41 + 2.23i)T \) |
| good | 11 | \( 1 - 5.65iT - 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 + 3.16iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 - 9.48iT - 61T^{2} \) |
| 67 | \( 1 - 7.07T + 67T^{2} \) |
| 71 | \( 1 - 1.41iT - 71T^{2} \) |
| 73 | \( 1 + 13.4iT - 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.872612594777860466448229736613, −7.80887521763118609529865125427, −7.36091875817717422224284166678, −6.77280187687909364754538423051, −5.95225964514081418774022328636, −4.95620486984806623481620775751, −4.39743541140931499597044712014, −3.46269411584201143085155757548, −2.35848901994209322197049451166, −0.880131520107419360271996422045,
0.52446932731773721536870436839, 1.97160056542526262419472543039, 2.75835059794702199864557875353, 3.65889971165393874279249114697, 4.40968964297318944912057985504, 5.52673997557790612487262959880, 6.14372434384362760483858262857, 6.81209830213457148975856283811, 8.128512679595718123253443913055, 8.627337828895685236093431065442
