| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s + 5·7-s − 3·8-s + 9-s + 6·11-s − 12-s − 3·13-s + 5·14-s − 16-s + 3·17-s + 18-s − 19-s + 5·21-s + 6·22-s + 5·23-s − 3·24-s − 3·26-s + 27-s − 5·28-s − 7·29-s + 5·32-s + 6·33-s + 3·34-s − 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.88·7-s − 1.06·8-s + 1/3·9-s + 1.80·11-s − 0.288·12-s − 0.832·13-s + 1.33·14-s − 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.229·19-s + 1.09·21-s + 1.27·22-s + 1.04·23-s − 0.612·24-s − 0.588·26-s + 0.192·27-s − 0.944·28-s − 1.29·29-s + 0.883·32-s + 1.04·33-s + 0.514·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.862704117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.862704117\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.662287083442801719642440565098, −7.83739020858218431400018637807, −7.22464364596293456405053938159, −6.19385836435589252683657042506, −5.24695523210207884231717135167, −4.69598239713040221591753257783, −4.04376656067980966343321857728, −3.27305892769118878756188860962, −2.02086459517443186217714462873, −1.13794858272902406205219014332,
1.13794858272902406205219014332, 2.02086459517443186217714462873, 3.27305892769118878756188860962, 4.04376656067980966343321857728, 4.69598239713040221591753257783, 5.24695523210207884231717135167, 6.19385836435589252683657042506, 7.22464364596293456405053938159, 7.83739020858218431400018637807, 8.662287083442801719642440565098
