| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s − 11-s − 2·13-s + 16-s + 6·17-s + 8·19-s + 2·20-s − 22-s − 4·23-s − 25-s − 2·26-s − 2·29-s − 8·31-s + 32-s + 6·34-s + 6·37-s + 8·38-s + 2·40-s + 6·41-s + 8·43-s − 44-s − 4·46-s + 4·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s − 0.301·11-s − 0.554·13-s + 1/4·16-s + 1.45·17-s + 1.83·19-s + 0.447·20-s − 0.213·22-s − 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.371·29-s − 1.43·31-s + 0.176·32-s + 1.02·34-s + 0.986·37-s + 1.29·38-s + 0.316·40-s + 0.937·41-s + 1.21·43-s − 0.150·44-s − 0.589·46-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.372475626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.372475626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.59354596545393433771081521175, −7.03723335031226019095847579745, −5.95216605049133194839739887642, −5.61405420269948436585325441601, −5.19457115960778750754121143472, −4.16177501065250447095868506483, −3.42542662388772626843959645805, −2.65949691733412216516782965779, −1.90012678727296365235754065158, −0.918528338634094893922527771013,
0.918528338634094893922527771013, 1.90012678727296365235754065158, 2.65949691733412216516782965779, 3.42542662388772626843959645805, 4.16177501065250447095868506483, 5.19457115960778750754121143472, 5.61405420269948436585325441601, 5.95216605049133194839739887642, 7.03723335031226019095847579745, 7.59354596545393433771081521175
