| L(s) = 1 | + 5-s + 7-s + 13-s − 17-s − 3·19-s + 6·23-s + 25-s − 3·29-s + 35-s − 5·37-s − 12·41-s + 8·43-s + 12·47-s − 6·49-s − 2·53-s + 65-s − 12·67-s + 71-s + 8·73-s + 8·79-s + 83-s − 85-s − 6·89-s + 91-s − 3·95-s − 8·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.277·13-s − 0.242·17-s − 0.688·19-s + 1.25·23-s + 1/5·25-s − 0.557·29-s + 0.169·35-s − 0.821·37-s − 1.87·41-s + 1.21·43-s + 1.75·47-s − 6/7·49-s − 0.274·53-s + 0.124·65-s − 1.46·67-s + 0.118·71-s + 0.936·73-s + 0.900·79-s + 0.109·83-s − 0.108·85-s − 0.635·89-s + 0.104·91-s − 0.307·95-s − 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.99293919877378, −13.72974630162084, −13.21129694624984, −12.65823326624971, −12.27515585970967, −11.66029920531903, −11.00632652539227, −10.78842756225877, −10.27418468371375, −9.589907238739776, −9.106112804555371, −8.676276780257344, −8.210089275674535, −7.505180730106287, −6.990898678002234, −6.540981150238878, −5.886492545810399, −5.370157443437192, −4.846335028537084, −4.268659357572851, −3.593289801234162, −2.973723108762860, −2.260135681832595, −1.678823515741975, −0.9917576499462806, 0,
0.9917576499462806, 1.678823515741975, 2.260135681832595, 2.973723108762860, 3.593289801234162, 4.268659357572851, 4.846335028537084, 5.370157443437192, 5.886492545810399, 6.540981150238878, 6.990898678002234, 7.505180730106287, 8.210089275674535, 8.676276780257344, 9.106112804555371, 9.589907238739776, 10.27418468371375, 10.78842756225877, 11.00632652539227, 11.66029920531903, 12.27515585970967, 12.65823326624971, 13.21129694624984, 13.72974630162084, 13.99293919877378
