| L(s) = 1 | − 2.79·2-s − 4.86·3-s − 0.165·4-s + 17.2·5-s + 13.6·6-s − 32.4·7-s + 22.8·8-s − 3.35·9-s − 48.2·10-s + 0.807·12-s + 78.3·13-s + 90.7·14-s − 83.8·15-s − 62.6·16-s − 17·17-s + 9.38·18-s + 90.7·19-s − 2.86·20-s + 157.·21-s − 44.1·23-s − 111.·24-s + 172.·25-s − 219.·26-s + 147.·27-s + 5.38·28-s − 98.3·29-s + 234.·30-s + ⋯ |
| L(s) = 1 | − 0.989·2-s − 0.935·3-s − 0.0207·4-s + 1.54·5-s + 0.926·6-s − 1.75·7-s + 1.01·8-s − 0.124·9-s − 1.52·10-s + 0.0194·12-s + 1.67·13-s + 1.73·14-s − 1.44·15-s − 0.978·16-s − 0.242·17-s + 0.122·18-s + 1.09·19-s − 0.0319·20-s + 1.63·21-s − 0.399·23-s − 0.945·24-s + 1.37·25-s − 1.65·26-s + 1.05·27-s + 0.0363·28-s − 0.630·29-s + 1.42·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8333894202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8333894202\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 + 2.79T + 8T^{2} \) |
| 3 | \( 1 + 4.86T + 27T^{2} \) |
| 5 | \( 1 - 17.2T + 125T^{2} \) |
| 7 | \( 1 + 32.4T + 343T^{2} \) |
| 13 | \( 1 - 78.3T + 2.19e3T^{2} \) |
| 19 | \( 1 - 90.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 44.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 98.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 262.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 106.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 394.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 281.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 239.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 517.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 545.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 219.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 111.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 250.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 553.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 75.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + 568.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 620.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.056883108834573218583372274527, −8.273616101759765785373881948395, −6.99458926774425652633710502074, −6.24357015707265003896878315445, −5.95610956996634242180999110595, −5.07488251495368203008866862383, −3.74254768484332021059066526721, −2.68063988524537565545363281444, −1.35359490505334858495201165311, −0.56938168459805135954434709453,
0.56938168459805135954434709453, 1.35359490505334858495201165311, 2.68063988524537565545363281444, 3.74254768484332021059066526721, 5.07488251495368203008866862383, 5.95610956996634242180999110595, 6.24357015707265003896878315445, 6.99458926774425652633710502074, 8.273616101759765785373881948395, 9.056883108834573218583372274527
