| L(s) = 1 | − 0.0401·3-s + (1.78 − 1.35i)5-s + 7-s − 2.99·9-s + 4.41i·11-s + 0.124i·13-s + (−0.0715 + 0.0543i)15-s + (−3.87 − 1.41i)17-s − 5.48·19-s − 0.0401·21-s − 5.88·23-s + (1.33 − 4.81i)25-s + 0.240·27-s + 1.55i·29-s − 7.94i·31-s + ⋯ |
| L(s) = 1 | − 0.0231·3-s + (0.796 − 0.605i)5-s + 0.377·7-s − 0.999·9-s + 1.33i·11-s + 0.0345i·13-s + (−0.0184 + 0.0140i)15-s + (−0.939 − 0.343i)17-s − 1.25·19-s − 0.00876·21-s − 1.22·23-s + (0.267 − 0.963i)25-s + 0.0463·27-s + 0.289i·29-s − 1.42i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4107054091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4107054091\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.78 + 1.35i)T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + (3.87 + 1.41i)T \) |
| good | 3 | \( 1 + 0.0401T + 3T^{2} \) |
| 11 | \( 1 - 4.41iT - 11T^{2} \) |
| 13 | \( 1 - 0.124iT - 13T^{2} \) |
| 19 | \( 1 + 5.48T + 19T^{2} \) |
| 23 | \( 1 + 5.88T + 23T^{2} \) |
| 29 | \( 1 - 1.55iT - 29T^{2} \) |
| 31 | \( 1 + 7.94iT - 31T^{2} \) |
| 37 | \( 1 + 8.67T + 37T^{2} \) |
| 41 | \( 1 + 12.2iT - 41T^{2} \) |
| 43 | \( 1 - 3.08iT - 43T^{2} \) |
| 47 | \( 1 + 4.44iT - 47T^{2} \) |
| 53 | \( 1 + 3.96iT - 53T^{2} \) |
| 59 | \( 1 + 5.88T + 59T^{2} \) |
| 61 | \( 1 + 5.75iT - 61T^{2} \) |
| 67 | \( 1 - 11.9iT - 67T^{2} \) |
| 71 | \( 1 + 13.3iT - 71T^{2} \) |
| 73 | \( 1 - 1.87T + 73T^{2} \) |
| 79 | \( 1 - 5.59iT - 79T^{2} \) |
| 83 | \( 1 - 8.93iT - 83T^{2} \) |
| 89 | \( 1 + 0.343T + 89T^{2} \) |
| 97 | \( 1 + 9.97T + 97T^{2} \) |
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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
−8.700222907609630543422760410579, −8.028091729625001447229161689211, −6.99671955012832832176137614876, −6.23331210161073667733714766090, −5.43946674705696659842491988224, −4.70300036051731860085069214525, −3.92334572894544543806687368199, −2.27189627142183354532544306137, −1.98533352882518147512286077763, −0.12194431732890236433654122037,
1.67234662235367650582836094140, 2.65030120490730826259374849929, 3.44723897697881607487984209158, 4.60032917056187729581092521482, 5.64193954025742678152394019534, 6.16952940862026144041704214887, 6.74734662205646972453298919516, 8.005791145799041092276187878667, 8.584116696194782933055598871912, 9.113461083793962453042963783827
