| L(s) = 1 | − 1.55·2-s − 3-s + 0.411·4-s − 3.25·5-s + 1.55·6-s + 1.50·7-s + 2.46·8-s + 9-s + 5.04·10-s + 2.02·11-s − 0.411·12-s + 0.242·13-s − 2.33·14-s + 3.25·15-s − 4.65·16-s + 17-s − 1.55·18-s − 6.76·19-s − 1.33·20-s − 1.50·21-s − 3.15·22-s + 1.35·23-s − 2.46·24-s + 5.56·25-s − 0.376·26-s − 27-s + 0.618·28-s + ⋯ |
| L(s) = 1 | − 1.09·2-s − 0.577·3-s + 0.205·4-s − 1.45·5-s + 0.633·6-s + 0.568·7-s + 0.872·8-s + 0.333·9-s + 1.59·10-s + 0.612·11-s − 0.118·12-s + 0.0673·13-s − 0.624·14-s + 0.839·15-s − 1.16·16-s + 0.242·17-s − 0.366·18-s − 1.55·19-s − 0.299·20-s − 0.328·21-s − 0.672·22-s + 0.281·23-s − 0.503·24-s + 1.11·25-s − 0.0739·26-s − 0.192·27-s + 0.116·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 1.55T + 2T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 7 | \( 1 - 1.50T + 7T^{2} \) |
| 11 | \( 1 - 2.02T + 11T^{2} \) |
| 13 | \( 1 - 0.242T + 13T^{2} \) |
| 19 | \( 1 + 6.76T + 19T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 29 | \( 1 + 3.55T + 29T^{2} \) |
| 31 | \( 1 + 2.72T + 31T^{2} \) |
| 37 | \( 1 - 1.05T + 37T^{2} \) |
| 41 | \( 1 - 9.95T + 41T^{2} \) |
| 43 | \( 1 + 4.11T + 43T^{2} \) |
| 47 | \( 1 + 4.69T + 47T^{2} \) |
| 53 | \( 1 - 7.08T + 53T^{2} \) |
| 59 | \( 1 - 9.72T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 7.81T + 67T^{2} \) |
| 71 | \( 1 + 0.214T + 71T^{2} \) |
| 73 | \( 1 - 2.44T + 73T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 5.24T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.323317853514118054914385902227, −7.40289344893880190044156537784, −7.02434872671584806264608545346, −5.99876187855071247359986070865, −4.88817801182976312496089619362, −4.26357443700934949236623336895, −3.66950946943758225961025512041, −2.07481654951611296309927565668, −0.979951056527419000133449471736, 0,
0.979951056527419000133449471736, 2.07481654951611296309927565668, 3.66950946943758225961025512041, 4.26357443700934949236623336895, 4.88817801182976312496089619362, 5.99876187855071247359986070865, 7.02434872671584806264608545346, 7.40289344893880190044156537784, 8.323317853514118054914385902227
