| L(s) = 1 | + 2-s − 4-s + 5-s − 2·7-s − 3·8-s + 10-s − 2·14-s − 16-s − 7·17-s + 6·19-s − 20-s + 6·23-s − 4·25-s + 2·28-s − 29-s + 4·31-s + 5·32-s − 7·34-s − 2·35-s + 37-s + 6·38-s − 3·40-s + 9·41-s − 6·43-s + 6·46-s − 6·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.755·7-s − 1.06·8-s + 0.316·10-s − 0.534·14-s − 1/4·16-s − 1.69·17-s + 1.37·19-s − 0.223·20-s + 1.25·23-s − 4/5·25-s + 0.377·28-s − 0.185·29-s + 0.718·31-s + 0.883·32-s − 1.20·34-s − 0.338·35-s + 0.164·37-s + 0.973·38-s − 0.474·40-s + 1.40·41-s − 0.914·43-s + 0.884·46-s − 0.875·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.38912922731199, −13.03032675809002, −12.71547130873099, −12.01017829435480, −11.42937953043864, −11.37412793505060, −10.40331227112696, −10.04971216363770, −9.516292899665899, −9.112240111820677, −8.840998885692751, −8.183086981812153, −7.485584414470569, −7.014048132764976, −6.349883973206672, −6.148052278577863, −5.454175089027601, −5.043859001179236, −4.484157605664782, −4.028346604156265, −3.341456499088553, −2.908163509065441, −2.415326205693917, −1.542666484978505, −0.7364318989111216, 0,
0.7364318989111216, 1.542666484978505, 2.415326205693917, 2.908163509065441, 3.341456499088553, 4.028346604156265, 4.484157605664782, 5.043859001179236, 5.454175089027601, 6.148052278577863, 6.349883973206672, 7.014048132764976, 7.485584414470569, 8.183086981812153, 8.840998885692751, 9.112240111820677, 9.516292899665899, 10.04971216363770, 10.40331227112696, 11.37412793505060, 11.42937953043864, 12.01017829435480, 12.71547130873099, 13.03032675809002, 13.38912922731199
