| L(s) = 1 | − 2.64·3-s + 3·5-s + 4.00·9-s − 2.64·11-s + 4·13-s − 7.93·15-s − 17-s + 7.93·19-s − 2.64·23-s + 4·25-s − 2.64·27-s − 4·29-s + 2.64·31-s + 7.00·33-s − 5·37-s − 10.5·39-s − 8·41-s + 10.5·43-s + 12.0·45-s − 2.64·47-s + 2.64·51-s + 7·53-s − 7.93·55-s − 21.0·57-s − 2.64·59-s + 5·61-s + 12·65-s + ⋯ |
| L(s) = 1 | − 1.52·3-s + 1.34·5-s + 1.33·9-s − 0.797·11-s + 1.10·13-s − 2.04·15-s − 0.242·17-s + 1.82·19-s − 0.551·23-s + 0.800·25-s − 0.509·27-s − 0.742·29-s + 0.475·31-s + 1.21·33-s − 0.821·37-s − 1.69·39-s − 1.24·41-s + 1.61·43-s + 1.78·45-s − 0.385·47-s + 0.370·51-s + 0.961·53-s − 1.07·55-s − 2.78·57-s − 0.344·59-s + 0.640·61-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.292563087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.292563087\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 - 7.93T + 19T^{2} \) |
| 23 | \( 1 + 2.64T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 2.64T + 47T^{2} \) |
| 53 | \( 1 - 7T + 53T^{2} \) |
| 59 | \( 1 + 2.64T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 2.64T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + 2.64T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−9.710317898047128283200313289802, −8.798742945798802303484967157152, −7.64765332138061986891050162184, −6.72254501150916638504420547699, −5.93469548582040074743719061273, −5.52933777388360428429996312991, −4.85647616507873890906603231389, −3.46065241215954745130870073412, −2.04206569018104086423028280813, −0.893580609759225263651882314677,
0.893580609759225263651882314677, 2.04206569018104086423028280813, 3.46065241215954745130870073412, 4.85647616507873890906603231389, 5.52933777388360428429996312991, 5.93469548582040074743719061273, 6.72254501150916638504420547699, 7.64765332138061986891050162184, 8.798742945798802303484967157152, 9.710317898047128283200313289802
