| L(s) = 1 | − 1.72·2-s − 3-s + 0.963·4-s − 5-s + 1.72·6-s + 1.78·8-s + 9-s + 1.72·10-s − 11-s − 0.963·12-s + 0.00287·13-s + 15-s − 4.99·16-s − 6.80·17-s − 1.72·18-s + 4.80·19-s − 0.963·20-s + 1.72·22-s + 3.96·23-s − 1.78·24-s + 25-s − 0.00495·26-s − 27-s − 8.94·29-s − 1.72·30-s + 9.39·31-s + 5.03·32-s + ⋯ |
| L(s) = 1 | − 1.21·2-s − 0.577·3-s + 0.481·4-s − 0.447·5-s + 0.702·6-s + 0.631·8-s + 0.333·9-s + 0.544·10-s − 0.301·11-s − 0.278·12-s + 0.000798·13-s + 0.258·15-s − 1.24·16-s − 1.65·17-s − 0.405·18-s + 1.10·19-s − 0.215·20-s + 0.366·22-s + 0.826·23-s − 0.364·24-s + 0.200·25-s − 0.000972·26-s − 0.192·27-s − 1.66·29-s − 0.314·30-s + 1.68·31-s + 0.889·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.72T + 2T^{2} \) |
| 13 | \( 1 - 0.00287T + 13T^{2} \) |
| 17 | \( 1 + 6.80T + 17T^{2} \) |
| 19 | \( 1 - 4.80T + 19T^{2} \) |
| 23 | \( 1 - 3.96T + 23T^{2} \) |
| 29 | \( 1 + 8.94T + 29T^{2} \) |
| 31 | \( 1 - 9.39T + 31T^{2} \) |
| 37 | \( 1 + 8.65T + 37T^{2} \) |
| 41 | \( 1 - 4.38T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 5.94T + 47T^{2} \) |
| 53 | \( 1 - 7.04T + 53T^{2} \) |
| 59 | \( 1 + 4.88T + 59T^{2} \) |
| 61 | \( 1 + 9.85T + 61T^{2} \) |
| 67 | \( 1 - 6.18T + 67T^{2} \) |
| 71 | \( 1 - 6.16T + 71T^{2} \) |
| 73 | \( 1 + 7.12T + 73T^{2} \) |
| 79 | \( 1 + 9.91T + 79T^{2} \) |
| 83 | \( 1 + 4.87T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−7.44388107097033991278740338765, −7.11635712671903953140629006595, −6.31778208920910627559537144069, −5.37867429669460113138391493652, −4.67586384629567123845831825065, −4.04964754100813524102765087632, −2.91077633656713026105354540250, −1.90216135126180703998128850597, −0.907607120391269110932490759083, 0,
0.907607120391269110932490759083, 1.90216135126180703998128850597, 2.91077633656713026105354540250, 4.04964754100813524102765087632, 4.67586384629567123845831825065, 5.37867429669460113138391493652, 6.31778208920910627559537144069, 7.11635712671903953140629006595, 7.44388107097033991278740338765
