| L(s) = 1 | + (0.0481 + 0.0944i)2-s + (1.16 − 1.60i)4-s + (−0.343 − 2.17i)5-s + (2.06 + 2.41i)7-s + (0.417 + 0.0661i)8-s + (0.188 − 0.136i)10-s + (0.444 − 0.725i)11-s + (0.161 + 2.05i)13-s + (−0.128 + 0.310i)14-s + (−1.21 − 3.74i)16-s + (−0.992 − 4.13i)17-s + (0.374 − 4.76i)19-s + (−3.89 − 1.98i)20-s + (0.0899 + 0.00707i)22-s + (0.549 − 1.69i)23-s + ⋯ |
| L(s) = 1 | + (0.0340 + 0.0667i)2-s + (0.584 − 0.804i)4-s + (−0.153 − 0.971i)5-s + (0.778 + 0.912i)7-s + (0.147 + 0.0233i)8-s + (0.0596 − 0.0433i)10-s + (0.134 − 0.218i)11-s + (0.0448 + 0.570i)13-s + (−0.0344 + 0.0830i)14-s + (−0.303 − 0.935i)16-s + (−0.240 − 1.00i)17-s + (0.0859 − 1.09i)19-s + (−0.871 − 0.443i)20-s + (0.0191 + 0.00150i)22-s + (0.114 − 0.352i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45743 - 0.705894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45743 - 0.705894i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 41 | \( 1 + (4.77 - 4.26i)T \) |
| good | 2 | \( 1 + (-0.0481 - 0.0944i)T + (-1.17 + 1.61i)T^{2} \) |
| 5 | \( 1 + (0.343 + 2.17i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (-2.06 - 2.41i)T + (-1.09 + 6.91i)T^{2} \) |
| 11 | \( 1 + (-0.444 + 0.725i)T + (-4.99 - 9.80i)T^{2} \) |
| 13 | \( 1 + (-0.161 - 2.05i)T + (-12.8 + 2.03i)T^{2} \) |
| 17 | \( 1 + (0.992 + 4.13i)T + (-15.1 + 7.71i)T^{2} \) |
| 19 | \( 1 + (-0.374 + 4.76i)T + (-18.7 - 2.97i)T^{2} \) |
| 23 | \( 1 + (-0.549 + 1.69i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (2.06 - 8.58i)T + (-25.8 - 13.1i)T^{2} \) |
| 31 | \( 1 + (-0.515 - 0.709i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.82 - 2.05i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (4.25 - 2.16i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-4.62 - 3.94i)T + (7.35 + 46.4i)T^{2} \) |
| 53 | \( 1 + (-9.67 - 2.32i)T + (47.2 + 24.0i)T^{2} \) |
| 59 | \( 1 + (5.78 + 1.88i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (6.59 - 12.9i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (-0.0572 - 0.0934i)T + (-30.4 + 59.6i)T^{2} \) |
| 71 | \( 1 + (-4.24 - 2.59i)T + (32.2 + 63.2i)T^{2} \) |
| 73 | \( 1 + (4.00 + 4.00i)T + 73iT^{2} \) |
| 79 | \( 1 + (5.12 + 12.3i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 8.82iT - 83T^{2} \) |
| 89 | \( 1 + (3.19 - 2.72i)T + (13.9 - 87.9i)T^{2} \) |
| 97 | \( 1 + (-1.80 + 1.10i)T + (44.0 - 86.4i)T^{2} \) |
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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
−11.51833947967367439634078518311, −10.46640577756270452302356858596, −9.081862053512220318934590913940, −8.853958749214461346195668616676, −7.44144444722599293128041331134, −6.40977710679684332170180863844, −5.18216227909489387732715634853, −4.75115813703667380018815662346, −2.62984913051743466428345935997, −1.26008119771988348281832073257,
1.95981979593265516709282732463, 3.41360529435019836158245103935, 4.20448402189242965361308189618, 5.92789608489069283883170732257, 7.01897698178990636417090482570, 7.69492781138336547626996683775, 8.397803914430679946931035004362, 10.07031644843881191341167628883, 10.74279295400825816332599536310, 11.40093004793666586350897937783
