| L(s) = 1 | − 3-s − 2·4-s − 5-s + 7-s − 2·9-s − 3·11-s + 2·12-s − 5·13-s + 15-s + 4·16-s + 3·17-s + 2·20-s − 21-s − 6·23-s + 25-s + 5·27-s − 2·28-s − 3·29-s + 4·31-s + 3·33-s − 35-s + 4·36-s − 2·37-s + 5·39-s + 12·41-s − 10·43-s + 6·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s + 0.727·17-s + 0.447·20-s − 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.377·28-s − 0.557·29-s + 0.718·31-s + 0.522·33-s − 0.169·35-s + 2/3·36-s − 0.328·37-s + 0.800·39-s + 1.87·41-s − 1.52·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−16.73833772606350, −16.07999318976877, −15.48879804047632, −14.59507147485527, −14.49886594548113, −13.83220515291904, −13.16116708915106, −12.36924537809527, −12.20903914769340, −11.53818690072543, −10.80472637385149, −10.24407977807533, −9.708730844030793, −9.075981835844921, −8.182802351905950, −7.937571190857328, −7.354492058876746, −6.303039086914539, −5.600891510660543, −5.105839934293347, −4.632497873715320, −3.817277033670284, −2.977970502058007, −2.148704702768943, −0.7527228293787584, 0,
0.7527228293787584, 2.148704702768943, 2.977970502058007, 3.817277033670284, 4.632497873715320, 5.105839934293347, 5.600891510660543, 6.303039086914539, 7.354492058876746, 7.937571190857328, 8.182802351905950, 9.075981835844921, 9.708730844030793, 10.24407977807533, 10.80472637385149, 11.53818690072543, 12.20903914769340, 12.36924537809527, 13.16116708915106, 13.83220515291904, 14.49886594548113, 14.59507147485527, 15.48879804047632, 16.07999318976877, 16.73833772606350
