L(s) = 1 | + (1.95 + 1.95i)2-s + (0.707 + 0.707i)3-s + 5.67i·4-s + (−2.22 − 0.224i)5-s + 2.76i·6-s + (−7.18 + 7.18i)8-s + 1.00i·9-s + (−3.91 − 4.79i)10-s + 3.09·11-s + (−4.00 + 4.00i)12-s + (1.01 + 1.01i)13-s + (−1.41 − 1.73i)15-s − 16.8·16-s + (2.64 − 2.64i)17-s + (−1.95 + 1.95i)18-s − 2.30·19-s + ⋯ |
L(s) = 1 | + (1.38 + 1.38i)2-s + (0.408 + 0.408i)3-s + 2.83i·4-s + (−0.994 − 0.100i)5-s + 1.13i·6-s + (−2.54 + 2.54i)8-s + 0.333i·9-s + (−1.23 − 1.51i)10-s + 0.933·11-s + (−1.15 + 1.15i)12-s + (0.280 + 0.280i)13-s + (−0.365 − 0.447i)15-s − 4.20·16-s + (0.641 − 0.641i)17-s + (−0.461 + 0.461i)18-s − 0.527·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.130560 - 2.92401i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.130560 - 2.92401i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (2.22 + 0.224i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.95 - 1.95i)T + 2iT^{2} \) |
| 11 | \( 1 - 3.09T + 11T^{2} \) |
| 13 | \( 1 + (-1.01 - 1.01i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.64 + 2.64i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.30T + 19T^{2} \) |
| 23 | \( 1 + (1.14 - 1.14i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.62iT - 29T^{2} \) |
| 31 | \( 1 + 0.287iT - 31T^{2} \) |
| 37 | \( 1 + (-1.96 - 1.96i)T + 37iT^{2} \) |
| 41 | \( 1 - 10.7iT - 41T^{2} \) |
| 43 | \( 1 + (1.32 - 1.32i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.72 + 2.72i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.49 + 2.49i)T - 53iT^{2} \) |
| 59 | \( 1 - 8.98T + 59T^{2} \) |
| 61 | \( 1 - 8.64iT - 61T^{2} \) |
| 67 | \( 1 + (-8.73 - 8.73i)T + 67iT^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + (-1.16 - 1.16i)T + 73iT^{2} \) |
| 79 | \( 1 + 7.82iT - 79T^{2} \) |
| 83 | \( 1 + (2.41 + 2.41i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.44T + 89T^{2} \) |
| 97 | \( 1 + (8.04 - 8.04i)T - 97iT^{2} \) |
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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
−11.40396464186381418472432173207, −9.660442526230551950346919152739, −8.600454009892828250238545588771, −8.075790151072450720181950926064, −7.19592687062798055051613366806, −6.44100003071461804755535742173, −5.37103185978541244037595420171, −4.32780356650236383023459001724, −3.89546048480594595910184967544, −2.86986847691468192248752939331,
0.994191196237968258314802429054, 2.28084443172177351251572547153, 3.59295275686299131252222271399, 3.86134489005081004223368905620, 5.08900791962491905176650034716, 6.19307123744491183698639198709, 7.03223040049935310548696591013, 8.394947277693773978329454389446, 9.294489631442924797327439799965, 10.38573937475140033086740896155