| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 3·11-s + 13-s + 15-s − 17-s + 19-s + 21-s + 6·23-s + 25-s + 27-s − 9·29-s + 2·31-s − 3·33-s + 35-s + 5·37-s + 39-s − 11·41-s − 2·43-s + 45-s − 3·47-s − 6·49-s − 51-s − 9·53-s − 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.258·15-s − 0.242·17-s + 0.229·19-s + 0.218·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.67·29-s + 0.359·31-s − 0.522·33-s + 0.169·35-s + 0.821·37-s + 0.160·39-s − 1.71·41-s − 0.304·43-s + 0.149·45-s − 0.437·47-s − 6/7·49-s − 0.140·51-s − 1.23·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.50084153153110, −14.94826981574140, −14.58397925531944, −13.91077480885483, −13.39222854977375, −12.96469276397878, −12.66032180473303, −11.63093365162617, −11.22253871344145, −10.73940823906467, −9.977477477840911, −9.626629211145830, −8.957387090075365, −8.404190718253448, −7.902812805213733, −7.297690345624236, −6.699163605934732, −6.014007828300190, −5.157515059804645, −4.960300400977462, −4.010517824169531, −3.256865242677317, −2.713975215153656, −1.899008276398790, −1.276532085326943, 0,
1.276532085326943, 1.899008276398790, 2.713975215153656, 3.256865242677317, 4.010517824169531, 4.960300400977462, 5.157515059804645, 6.014007828300190, 6.699163605934732, 7.297690345624236, 7.902812805213733, 8.404190718253448, 8.957387090075365, 9.626629211145830, 9.977477477840911, 10.73940823906467, 11.22253871344145, 11.63093365162617, 12.66032180473303, 12.96469276397878, 13.39222854977375, 13.91077480885483, 14.58397925531944, 14.94826981574140, 15.50084153153110
