| L(s) = 1 | + 3.79·2-s − 3.62·3-s + 6.43·4-s − 1.22·5-s − 13.7·6-s − 33.4·7-s − 5.93·8-s − 13.8·9-s − 4.66·10-s − 33.6·11-s − 23.3·12-s − 20.9·13-s − 127.·14-s + 4.45·15-s − 74.0·16-s + 84.6·17-s − 52.6·18-s − 40.5·19-s − 7.90·20-s + 121.·21-s − 127.·22-s − 67.6·23-s + 21.5·24-s − 123.·25-s − 79.5·26-s + 148.·27-s − 215.·28-s + ⋯ |
| L(s) = 1 | + 1.34·2-s − 0.697·3-s + 0.804·4-s − 0.109·5-s − 0.937·6-s − 1.80·7-s − 0.262·8-s − 0.513·9-s − 0.147·10-s − 0.922·11-s − 0.561·12-s − 0.446·13-s − 2.42·14-s + 0.0766·15-s − 1.15·16-s + 1.20·17-s − 0.689·18-s − 0.489·19-s − 0.0883·20-s + 1.26·21-s − 1.23·22-s − 0.613·23-s + 0.182·24-s − 0.987·25-s − 0.599·26-s + 1.05·27-s − 1.45·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3758876099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3758876099\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 3.79T + 8T^{2} \) |
| 3 | \( 1 + 3.62T + 27T^{2} \) |
| 5 | \( 1 + 1.22T + 125T^{2} \) |
| 7 | \( 1 + 33.4T + 343T^{2} \) |
| 11 | \( 1 + 33.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 40.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 67.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 176.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 89.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 147.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 216.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 453.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 285.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 810.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 901.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 707.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 849.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 179.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 554.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.47e3T + 9.12e5T^{2} \) |
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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
−9.027987327347907249642669826465, −7.85668095284822977670147065625, −6.94119307771114923136540953670, −6.02934436100576059339914660887, −5.76859661456916768589832594520, −4.97598927653759091830504535700, −3.81398847567630528491018394006, −3.22739766217802719097307983489, −2.37493779039270468969132167772, −0.22079917053153783322207183017,
0.22079917053153783322207183017, 2.37493779039270468969132167772, 3.22739766217802719097307983489, 3.81398847567630528491018394006, 4.97598927653759091830504535700, 5.76859661456916768589832594520, 6.02934436100576059339914660887, 6.94119307771114923136540953670, 7.85668095284822977670147065625, 9.027987327347907249642669826465
