| L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s + 10-s + 11-s + 2·13-s − 16-s + 6·17-s + 19-s + 20-s − 22-s + 8·23-s + 25-s − 2·26-s + 6·29-s + 4·31-s − 5·32-s − 6·34-s − 2·37-s − 38-s − 3·40-s + 10·41-s + 4·43-s − 44-s − 8·46-s − 7·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s + 0.316·10-s + 0.301·11-s + 0.554·13-s − 1/4·16-s + 1.45·17-s + 0.229·19-s + 0.223·20-s − 0.213·22-s + 1.66·23-s + 1/5·25-s − 0.392·26-s + 1.11·29-s + 0.718·31-s − 0.883·32-s − 1.02·34-s − 0.328·37-s − 0.162·38-s − 0.474·40-s + 1.56·41-s + 0.609·43-s − 0.150·44-s − 1.17·46-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.477230617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.477230617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.80887232048572540682333420947, −7.22327720234686822026420079539, −6.52535200691129039267664712399, −5.53828988498538538917364873372, −4.96830657216051678599610525118, −4.15724232120626547947827499344, −3.49732228870449175323239470061, −2.64373224217327831161992119768, −1.19373409267433679159515614244, −0.823941476270948826303287224898,
0.823941476270948826303287224898, 1.19373409267433679159515614244, 2.64373224217327831161992119768, 3.49732228870449175323239470061, 4.15724232120626547947827499344, 4.96830657216051678599610525118, 5.53828988498538538917364873372, 6.52535200691129039267664712399, 7.22327720234686822026420079539, 7.80887232048572540682333420947
