L(s) = 1 | − 0.902·2-s − 9.58·3-s − 7.18·4-s − 2.03·5-s + 8.65·6-s + 26.8·7-s + 13.7·8-s + 64.8·9-s + 1.83·10-s + 38.6·11-s + 68.8·12-s − 37.4·13-s − 24.2·14-s + 19.4·15-s + 45.1·16-s − 12.0·17-s − 58.5·18-s − 64.0·19-s + 14.6·20-s − 257.·21-s − 34.9·22-s − 85.2·23-s − 131.·24-s − 120.·25-s + 33.8·26-s − 362.·27-s − 192.·28-s + ⋯ |
L(s) = 1 | − 0.319·2-s − 1.84·3-s − 0.898·4-s − 0.181·5-s + 0.588·6-s + 1.44·7-s + 0.605·8-s + 2.40·9-s + 0.0580·10-s + 1.06·11-s + 1.65·12-s − 0.799·13-s − 0.462·14-s + 0.335·15-s + 0.704·16-s − 0.172·17-s − 0.766·18-s − 0.773·19-s + 0.163·20-s − 2.67·21-s − 0.338·22-s − 0.773·23-s − 1.11·24-s − 0.966·25-s + 0.255·26-s − 2.58·27-s − 1.30·28-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
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 0.902T + 8T^{2} \) |
| 3 | \( 1 + 9.58T + 27T^{2} \) |
| 5 | \( 1 + 2.03T + 125T^{2} \) |
| 7 | \( 1 - 26.8T + 343T^{2} \) |
| 11 | \( 1 - 38.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 12.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 64.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 85.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 129.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 261.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 310.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 333.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 475.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 14.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 652.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 233.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 963.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 887.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 345.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 396.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 539.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 295.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 191.T + 9.12e5T^{2} \) |
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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
−8.293744457501965285942258014914, −7.85421422104862583444943772344, −6.73541464155212468829890634690, −6.06780256284627150397582310962, −5.02849149094558640105812586546, −4.61460436890054248943260895693, −4.03775548988269611213185283699, −1.81584072821086226435019661451, −0.976167151364885328108151243550, 0,
0.976167151364885328108151243550, 1.81584072821086226435019661451, 4.03775548988269611213185283699, 4.61460436890054248943260895693, 5.02849149094558640105812586546, 6.06780256284627150397582310962, 6.73541464155212468829890634690, 7.85421422104862583444943772344, 8.293744457501965285942258014914