| L(s) = 1 | − 0.610·2-s + 1.93·3-s − 1.62·4-s − 1.75·5-s − 1.17·6-s + 2.21·8-s + 0.726·9-s + 1.07·10-s + 11-s − 3.14·12-s − 13-s − 3.38·15-s + 1.90·16-s + 6.96·17-s − 0.443·18-s − 3.17·19-s + 2.85·20-s − 0.610·22-s − 7.45·23-s + 4.27·24-s − 1.91·25-s + 0.610·26-s − 4.38·27-s + 8.40·29-s + 2.06·30-s + 1.51·31-s − 5.59·32-s + ⋯ |
| L(s) = 1 | − 0.431·2-s + 1.11·3-s − 0.813·4-s − 0.785·5-s − 0.481·6-s + 0.783·8-s + 0.242·9-s + 0.339·10-s + 0.301·11-s − 0.906·12-s − 0.277·13-s − 0.875·15-s + 0.475·16-s + 1.68·17-s − 0.104·18-s − 0.728·19-s + 0.638·20-s − 0.130·22-s − 1.55·23-s + 0.872·24-s − 0.383·25-s + 0.119·26-s − 0.844·27-s + 1.56·29-s + 0.377·30-s + 0.272·31-s − 0.988·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.610T + 2T^{2} \) |
| 3 | \( 1 - 1.93T + 3T^{2} \) |
| 5 | \( 1 + 1.75T + 5T^{2} \) |
| 17 | \( 1 - 6.96T + 17T^{2} \) |
| 19 | \( 1 + 3.17T + 19T^{2} \) |
| 23 | \( 1 + 7.45T + 23T^{2} \) |
| 29 | \( 1 - 8.40T + 29T^{2} \) |
| 31 | \( 1 - 1.51T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 + 2.05T + 41T^{2} \) |
| 43 | \( 1 - 8.28T + 43T^{2} \) |
| 47 | \( 1 + 7.74T + 47T^{2} \) |
| 53 | \( 1 - 9.67T + 53T^{2} \) |
| 59 | \( 1 + 9.22T + 59T^{2} \) |
| 61 | \( 1 - 4.81T + 61T^{2} \) |
| 67 | \( 1 + 3.73T + 67T^{2} \) |
| 71 | \( 1 + 0.781T + 71T^{2} \) |
| 73 | \( 1 - 0.513T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 4.71T + 83T^{2} \) |
| 89 | \( 1 - 4.72T + 89T^{2} \) |
| 97 | \( 1 + 9.98T + 97T^{2} \) |
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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
−7.974545653893336393489051309993, −7.33717104867770945930182070650, −6.24055699286588843727212559113, −5.41687232091187579354058918348, −4.44818958721390456538564764795, −3.88744675589121920948935275854, −3.29592584461472833053655509089, −2.31622650813057511126814244527, −1.20500236855286782788658943424, 0,
1.20500236855286782788658943424, 2.31622650813057511126814244527, 3.29592584461472833053655509089, 3.88744675589121920948935275854, 4.44818958721390456538564764795, 5.41687232091187579354058918348, 6.24055699286588843727212559113, 7.33717104867770945930182070650, 7.974545653893336393489051309993
