| L(s) = 1 | + (−2.23 − 0.0743i)5-s − 2.82i·7-s + 6.46·11-s − i·13-s + 3.49i·17-s + 2.21·19-s + 7.14i·23-s + (4.98 + 0.332i)25-s + 4.82·29-s − 9.65·31-s + (−0.210 + 6.32i)35-s − 6.93i·37-s + 0.148·41-s + 3.03i·43-s + 6.79i·47-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0332i)5-s − 1.06i·7-s + 1.95·11-s − 0.277i·13-s + 0.847i·17-s + 0.507·19-s + 1.49i·23-s + (0.997 + 0.0664i)25-s + 0.896·29-s − 1.73·31-s + (−0.0355 + 1.06i)35-s − 1.14i·37-s + 0.0232·41-s + 0.463i·43-s + 0.990i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.756743323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.756743323\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.23 + 0.0743i)T \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 6.46T + 11T^{2} \) |
| 17 | \( 1 - 3.49iT - 17T^{2} \) |
| 19 | \( 1 - 2.21T + 19T^{2} \) |
| 23 | \( 1 - 7.14iT - 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + 9.65T + 31T^{2} \) |
| 37 | \( 1 + 6.93iT - 37T^{2} \) |
| 41 | \( 1 - 0.148T + 41T^{2} \) |
| 43 | \( 1 - 3.03iT - 43T^{2} \) |
| 47 | \( 1 - 6.79iT - 47T^{2} \) |
| 53 | \( 1 + 1.70iT - 53T^{2} \) |
| 59 | \( 1 + 6.46T + 59T^{2} \) |
| 61 | \( 1 - 2.66T + 61T^{2} \) |
| 67 | \( 1 - 7.70iT - 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 5.76iT - 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 2.27iT - 83T^{2} \) |
| 89 | \( 1 + 9.21T + 89T^{2} \) |
| 97 | \( 1 - 8.11iT - 97T^{2} \) |
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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
−8.189699678894423978412858836665, −7.47372757029133111399519880434, −7.03792553907879381474136087487, −6.26510735608430400273665514995, −5.33359615010690816900973603605, −4.21495409552506230372776483408, −3.86773891638880212649838770164, −3.27485520857114304492958276150, −1.61131556374661327996093130279, −0.844041046199045154512080281902,
0.70608390833137749883023924741, 1.93595249317382685822346434748, 3.01516567234332177254727604518, 3.75563336384876591584690641216, 4.55414850867629854292898961285, 5.25946062588027449696022865059, 6.38073339795118088355790783032, 6.74334119080248333264726039325, 7.56437397795944795042544553599, 8.484228062857975780060022286154
