| L(s) = 1 | − 3.47e4i·2-s − 4.98e7i·3-s + 7.38e9·4-s − 1.73e12·6-s − 1.47e14i·7-s − 5.55e14i·8-s + 3.07e15·9-s + 5.98e16·11-s − 3.67e17i·12-s + 1.87e18i·13-s − 5.13e18·14-s + 4.41e19·16-s − 1.50e19i·17-s − 1.06e20i·18-s − 9.64e20·19-s + ⋯ |
| L(s) = 1 | − 0.374i·2-s − 0.668i·3-s + 0.859·4-s − 0.250·6-s − 1.68i·7-s − 0.697i·8-s + 0.553·9-s + 0.392·11-s − 0.574i·12-s + 0.780i·13-s − 0.630·14-s + 0.598·16-s − 0.0748i·17-s − 0.207i·18-s − 0.767·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(\approx\) |
\(2.257638377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.257638377\) |
| \(L(\frac{35}{2})\) |
|
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 | \( 1 + 3.47e4iT - 8.58e9T^{2} \) |
| 3 | \( 1 + 4.98e7iT - 5.55e15T^{2} \) |
| 7 | \( 1 + 1.47e14iT - 7.73e27T^{2} \) |
| 11 | \( 1 - 5.98e16T + 2.32e34T^{2} \) |
| 13 | \( 1 - 1.87e18iT - 5.75e36T^{2} \) |
| 17 | \( 1 + 1.50e19iT - 4.02e40T^{2} \) |
| 19 | \( 1 + 9.64e20T + 1.58e42T^{2} \) |
| 23 | \( 1 + 2.94e22iT - 8.65e44T^{2} \) |
| 29 | \( 1 + 2.20e24T + 1.81e48T^{2} \) |
| 31 | \( 1 - 2.16e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 1.14e26iT - 5.63e51T^{2} \) |
| 41 | \( 1 + 5.38e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 5.52e25iT - 8.02e53T^{2} \) |
| 47 | \( 1 - 2.40e26iT - 1.51e55T^{2} \) |
| 53 | \( 1 + 3.29e28iT - 7.96e56T^{2} \) |
| 59 | \( 1 - 2.15e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 2.06e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 2.34e30iT - 1.82e60T^{2} \) |
| 71 | \( 1 - 5.77e29T + 1.23e61T^{2} \) |
| 73 | \( 1 + 1.07e30iT - 3.08e61T^{2} \) |
| 79 | \( 1 - 2.26e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 2.41e30iT - 2.13e63T^{2} \) |
| 89 | \( 1 + 2.76e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 1.55e32iT - 3.65e65T^{2} \) |
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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
−10.70774015834561715117101424201, −9.837845651576448168153768076362, −7.992331567166989743775485379670, −6.88887752367545670725240601564, −6.58465405846650894916903964915, −4.44537699761340815952123739972, −3.54711104617067493493251202122, −2.02021539318566573704130453991, −1.33017835049329723163232606629, −0.36787134876490534162540625521,
1.58196082069455006474640672358, 2.50573900009377791749172435398, 3.69279039921929087474665382542, 5.27220788136320385110533194359, 5.94644302576003927296495400223, 7.26097945248739797517766310405, 8.523902709672728728825646485867, 9.584829878520679572804184708499, 10.83221515614158838884694183686, 11.86509873553561634816007461293
