| L(s) = 1 | + (−1.55 + 1.60i)5-s − 1.21·7-s + 2.73·11-s + 6.99i·13-s + 0.391i·17-s − 2.94i·19-s + (3.74 + 3.00i)23-s + (−0.138 − 4.99i)25-s − 3.30i·29-s + 7.96·31-s + (1.89 − 1.94i)35-s − 1.84·37-s + 9.35i·41-s + 4.89·43-s + 2.87·47-s + ⋯ |
| L(s) = 1 | + (−0.697 + 0.716i)5-s − 0.459·7-s + 0.825·11-s + 1.94i·13-s + 0.0948i·17-s − 0.675i·19-s + (0.779 + 0.625i)23-s + (−0.0277 − 0.999i)25-s − 0.613i·29-s + 1.42·31-s + (0.320 − 0.329i)35-s − 0.303·37-s + 1.46i·41-s + 0.747·43-s + 0.419·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.187128974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.187128974\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.55 - 1.60i)T \) |
| 23 | \( 1 + (-3.74 - 3.00i)T \) |
| good | 7 | \( 1 + 1.21T + 7T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 13 | \( 1 - 6.99iT - 13T^{2} \) |
| 17 | \( 1 - 0.391iT - 17T^{2} \) |
| 19 | \( 1 + 2.94iT - 19T^{2} \) |
| 29 | \( 1 + 3.30iT - 29T^{2} \) |
| 31 | \( 1 - 7.96T + 31T^{2} \) |
| 37 | \( 1 + 1.84T + 37T^{2} \) |
| 41 | \( 1 - 9.35iT - 41T^{2} \) |
| 43 | \( 1 - 4.89T + 43T^{2} \) |
| 47 | \( 1 - 2.87T + 47T^{2} \) |
| 53 | \( 1 - 2.95iT - 53T^{2} \) |
| 59 | \( 1 - 8.16iT - 59T^{2} \) |
| 61 | \( 1 + 12.9iT - 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 14.5iT - 71T^{2} \) |
| 73 | \( 1 + 1.70iT - 73T^{2} \) |
| 79 | \( 1 + 11.9iT - 79T^{2} \) |
| 83 | \( 1 - 12.9iT - 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.765968093152827296274805953250, −7.88566175853406978111614676034, −6.99906888411197433562752458619, −6.67644608653862675322527756891, −6.02050614980757188770011853677, −4.61802968736298762532108558505, −4.20920265795482293801392726170, −3.29401448087530618269413754262, −2.46977649371945788908003920227, −1.23408417709062529557075071305,
0.39190700330847519018618713108, 1.28550603480595076304782137134, 2.82332466051224635110302078072, 3.49976646083319415533876408835, 4.32353570911112204796799842267, 5.18455499511240939291396030869, 5.84435452396816684121214012737, 6.73838050191200715166209329872, 7.52679097609173831665355225476, 8.197670368549549705324004861065
