| L(s) = 1 | − 3.98·2-s − 2.76·3-s + 7.90·4-s − 14.0·5-s + 11.0·6-s + 21.9·7-s + 0.384·8-s − 19.3·9-s + 55.8·10-s − 21.8·12-s − 13·13-s − 87.4·14-s + 38.7·15-s − 64.7·16-s + 72.0·17-s + 77.1·18-s + 87.9·19-s − 110.·20-s − 60.6·21-s − 7.78·23-s − 1.06·24-s + 71.1·25-s + 51.8·26-s + 128.·27-s + 173.·28-s − 59.3·29-s − 154.·30-s + ⋯ |
| L(s) = 1 | − 1.40·2-s − 0.532·3-s + 0.987·4-s − 1.25·5-s + 0.750·6-s + 1.18·7-s + 0.0170·8-s − 0.716·9-s + 1.76·10-s − 0.525·12-s − 0.277·13-s − 1.66·14-s + 0.666·15-s − 1.01·16-s + 1.02·17-s + 1.01·18-s + 1.06·19-s − 1.23·20-s − 0.630·21-s − 0.0705·23-s − 0.00905·24-s + 0.569·25-s + 0.391·26-s + 0.913·27-s + 1.16·28-s − 0.380·29-s − 0.940·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5556803293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5556803293\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 3.98T + 8T^{2} \) |
| 3 | \( 1 + 2.76T + 27T^{2} \) |
| 5 | \( 1 + 14.0T + 125T^{2} \) |
| 7 | \( 1 - 21.9T + 343T^{2} \) |
| 17 | \( 1 - 72.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 87.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.78T + 1.21e4T^{2} \) |
| 29 | \( 1 + 59.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 69.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 146.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 465.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 156.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 89.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 237.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 756.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 665.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 706.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 93.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + 790.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 226.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 911.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 936.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.06e3T + 9.12e5T^{2} \) |
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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
−8.900698936300370084666324585902, −8.213476952239757656667309217261, −7.65207510988374146319778993358, −7.22305397552755769830107876147, −5.82174455158705139914912776322, −4.97710010708460888186238106257, −4.05726611396721839379696327674, −2.76740074360870661361022849274, −1.34622981844017857515071744170, −0.51638931012055450747624000364,
0.51638931012055450747624000364, 1.34622981844017857515071744170, 2.76740074360870661361022849274, 4.05726611396721839379696327674, 4.97710010708460888186238106257, 5.82174455158705139914912776322, 7.22305397552755769830107876147, 7.65207510988374146319778993358, 8.213476952239757656667309217261, 8.900698936300370084666324585902
