| L(s) = 1 | − 4.56·2-s + 12.8·4-s + 2.80·5-s − 9.56·7-s − 21.9·8-s − 12.8·10-s + 39.4·11-s + 43.6·14-s − 2.42·16-s − 2.01·17-s + 60.1·19-s + 35.9·20-s − 179.·22-s − 4.46·23-s − 117.·25-s − 122.·28-s − 140.·29-s − 136.·31-s + 186.·32-s + 9.19·34-s − 26.8·35-s + 185.·37-s − 274.·38-s − 61.5·40-s + 310.·41-s + 427.·43-s + 504.·44-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.60·4-s + 0.251·5-s − 0.516·7-s − 0.969·8-s − 0.405·10-s + 1.08·11-s + 0.832·14-s − 0.0378·16-s − 0.0287·17-s + 0.726·19-s + 0.402·20-s − 1.74·22-s − 0.0405·23-s − 0.936·25-s − 0.826·28-s − 0.900·29-s − 0.788·31-s + 1.03·32-s + 0.0463·34-s − 0.129·35-s + 0.825·37-s − 1.17·38-s − 0.243·40-s + 1.18·41-s + 1.51·43-s + 1.73·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.56T + 8T^{2} \) |
| 5 | \( 1 - 2.80T + 125T^{2} \) |
| 7 | \( 1 + 9.56T + 343T^{2} \) |
| 11 | \( 1 - 39.4T + 1.33e3T^{2} \) |
| 17 | \( 1 + 2.01T + 4.91e3T^{2} \) |
| 19 | \( 1 - 60.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 4.46T + 1.21e4T^{2} \) |
| 29 | \( 1 + 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 136.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 185.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 310.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 427.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 258.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 612.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 517.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 161.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 49.8T + 3.00e5T^{2} \) |
| 71 | \( 1 - 279.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 467.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 37.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 76.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 202.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.17e3T + 9.12e5T^{2} \) |
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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
−9.028846860301285114815939076465, −7.895243051377449855482698446461, −7.40662131209223513604549297313, −6.45576384966031824968973947886, −5.82723221069999670170110774503, −4.37589210240560850595647254091, −3.25886754824020089737239112160, −2.03329052862266446654586087612, −1.14480537131054448136326617302, 0,
1.14480537131054448136326617302, 2.03329052862266446654586087612, 3.25886754824020089737239112160, 4.37589210240560850595647254091, 5.82723221069999670170110774503, 6.45576384966031824968973947886, 7.40662131209223513604549297313, 7.895243051377449855482698446461, 9.028846860301285114815939076465
