| L(s) = 1 | + 3-s + 3·5-s + 7-s − 2·9-s + 6·11-s − 13-s + 3·15-s − 3·17-s + 2·19-s + 21-s + 4·25-s − 5·27-s − 6·29-s + 4·31-s + 6·33-s + 3·35-s + 7·37-s − 39-s − 43-s − 6·45-s − 3·47-s − 6·49-s − 3·51-s + 18·55-s + 2·57-s − 6·59-s − 8·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 0.377·7-s − 2/3·9-s + 1.80·11-s − 0.277·13-s + 0.774·15-s − 0.727·17-s + 0.458·19-s + 0.218·21-s + 4/5·25-s − 0.962·27-s − 1.11·29-s + 0.718·31-s + 1.04·33-s + 0.507·35-s + 1.15·37-s − 0.160·39-s − 0.152·43-s − 0.894·45-s − 0.437·47-s − 6/7·49-s − 0.420·51-s + 2.42·55-s + 0.264·57-s − 0.781·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.460267371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.460267371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.874010062860047513080661907582, −9.336106539144567801750341824832, −8.795157144430056998341032474834, −7.75239123229149118292385279471, −6.55807134346327510973080228663, −5.98996247421716782722424041623, −4.89430736538204861389445208248, −3.70239520646251101168283863338, −2.47134717318492855260086903538, −1.50897905814590959854792778451,
1.50897905814590959854792778451, 2.47134717318492855260086903538, 3.70239520646251101168283863338, 4.89430736538204861389445208248, 5.98996247421716782722424041623, 6.55807134346327510973080228663, 7.75239123229149118292385279471, 8.795157144430056998341032474834, 9.336106539144567801750341824832, 9.874010062860047513080661907582
