| L(s) = 1 | − 1.53·3-s − 5-s − 5.03·7-s − 0.633·9-s + 3.03·11-s − 4.57·13-s + 1.53·15-s − 1.07·17-s − 19-s + 7.74·21-s − 4.11·23-s + 25-s + 5.58·27-s − 1.07·29-s − 5.58·31-s − 4.66·33-s + 5.03·35-s + 0.0947·37-s + 7.03·39-s + 10.6·41-s − 5.03·43-s + 0.633·45-s + 12.2·47-s + 18.3·49-s + 1.65·51-s − 4.09·53-s − 3.03·55-s + ⋯ |
| L(s) = 1 | − 0.888·3-s − 0.447·5-s − 1.90·7-s − 0.211·9-s + 0.914·11-s − 1.26·13-s + 0.397·15-s − 0.261·17-s − 0.229·19-s + 1.68·21-s − 0.857·23-s + 0.200·25-s + 1.07·27-s − 0.199·29-s − 1.00·31-s − 0.812·33-s + 0.850·35-s + 0.0155·37-s + 1.12·39-s + 1.66·41-s − 0.767·43-s + 0.0943·45-s + 1.79·47-s + 2.61·49-s + 0.231·51-s − 0.562·53-s − 0.408·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4262756366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4262756366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.53T + 3T^{2} \) |
| 7 | \( 1 + 5.03T + 7T^{2} \) |
| 11 | \( 1 - 3.03T + 11T^{2} \) |
| 13 | \( 1 + 4.57T + 13T^{2} \) |
| 17 | \( 1 + 1.07T + 17T^{2} \) |
| 23 | \( 1 + 4.11T + 23T^{2} \) |
| 29 | \( 1 + 1.07T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 - 0.0947T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 5.03T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + 4.09T + 53T^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 + 5.69T + 61T^{2} \) |
| 67 | \( 1 + 5.28T + 67T^{2} \) |
| 71 | \( 1 - 5.67T + 71T^{2} \) |
| 73 | \( 1 - 9.07T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 + 1.95T + 83T^{2} \) |
| 89 | \( 1 + 2.18T + 89T^{2} \) |
| 97 | \( 1 + 2.16T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.449430027250702613786232117533, −8.964039461575844895619044848600, −7.63641265662152737540397743175, −6.85577843701609289855281673901, −6.24411125598130606474362460943, −5.55465297024890703497978560209, −4.36632568048107215959714918484, −3.52045130597605910564229139364, −2.46917309036851989321327263069, −0.45241347681416962754148785768,
0.45241347681416962754148785768, 2.46917309036851989321327263069, 3.52045130597605910564229139364, 4.36632568048107215959714918484, 5.55465297024890703497978560209, 6.24411125598130606474362460943, 6.85577843701609289855281673901, 7.63641265662152737540397743175, 8.964039461575844895619044848600, 9.449430027250702613786232117533
