| L(s) = 1 | + 1.32i·2-s − 2.14i·3-s + 0.236·4-s + 2.85·6-s − 3.47i·7-s + 2.96i·8-s − 1.61·9-s + 2·11-s − 0.507i·12-s + 2.65i·13-s + 4.61·14-s − 3.47·16-s + 4.29i·17-s − 2.14i·18-s − 7.23·19-s + ⋯ |
| L(s) = 1 | + 0.939i·2-s − 1.24i·3-s + 0.118·4-s + 1.16·6-s − 1.31i·7-s + 1.04i·8-s − 0.539·9-s + 0.603·11-s − 0.146i·12-s + 0.736i·13-s + 1.23·14-s − 0.868·16-s + 1.04i·17-s − 0.506i·18-s − 1.66·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18391\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 1.32iT - 2T^{2} \) |
| 3 | \( 1 + 2.14iT - 3T^{2} \) |
| 7 | \( 1 + 3.47iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2.65iT - 13T^{2} \) |
| 17 | \( 1 - 4.29iT - 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 - 0.820iT - 23T^{2} \) |
| 29 | \( 1 + 0.854T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 1.64iT - 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 - 3.79iT - 43T^{2} \) |
| 47 | \( 1 + 0.507iT - 47T^{2} \) |
| 53 | \( 1 - 8.59iT - 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 - 5.09T + 61T^{2} \) |
| 67 | \( 1 - 4.29iT - 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 + 16.5iT - 73T^{2} \) |
| 79 | \( 1 + 2.76T + 79T^{2} \) |
| 83 | \( 1 + 5.11iT - 83T^{2} \) |
| 89 | \( 1 - 8.61T + 89T^{2} \) |
| 97 | \( 1 + 11.2iT - 97T^{2} \) |
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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
−13.55824069596248487525049594053, −12.59383178982824074029754414680, −11.45225317064781691469442805310, −10.41100051466200062276639691523, −8.619722164361711454286662530612, −7.66052471860732777142233312614, −6.78812819921853672503410232780, −6.24265964925761439474038756120, −4.23868086200875547629532658995, −1.78477214729598544163120545430,
2.46155450755188348858761391626, 3.77899383550426139093334925518, 5.15978931484839124999184725875, 6.60582569533073179545512266510, 8.571413793792958665948500543945, 9.512019967162454171782597200886, 10.32784007284325006451767032209, 11.27866847001145691181002116767, 12.10796884819604239734538664712, 13.03426254283569927531447358420
