| L(s) = 1 | − 2·3-s − 5-s + 9-s + 2·11-s − 13-s + 2·15-s − 2·17-s − 2·19-s + 2·23-s + 25-s + 4·27-s − 6·29-s − 2·31-s − 4·33-s − 6·37-s + 2·39-s − 2·41-s + 6·43-s − 45-s + 8·47-s + 4·51-s − 2·53-s − 2·55-s + 4·57-s − 6·59-s + 14·61-s + 65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.516·15-s − 0.485·17-s − 0.458·19-s + 0.417·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s − 0.359·31-s − 0.696·33-s − 0.986·37-s + 0.320·39-s − 0.312·41-s + 0.914·43-s − 0.149·45-s + 1.16·47-s + 0.560·51-s − 0.274·53-s − 0.269·55-s + 0.529·57-s − 0.781·59-s + 1.79·61-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.67553066579770, −15.21097736642868, −14.48214839860664, −14.18122894341237, −13.31033069627858, −12.78712154160166, −12.25799946675980, −11.85216296655959, −11.19743665250933, −10.95012506168618, −10.38148636378456, −9.621873520805828, −9.002944611147541, −8.538267681205997, −7.745619820771932, −7.023040014928045, −6.732798356826361, −5.952312981071799, −5.455657562316629, −4.849632717528888, −4.153806448983733, −3.610005636350079, −2.650664392039184, −1.789540118610227, −0.8067294653983284, 0,
0.8067294653983284, 1.789540118610227, 2.650664392039184, 3.610005636350079, 4.153806448983733, 4.849632717528888, 5.455657562316629, 5.952312981071799, 6.732798356826361, 7.023040014928045, 7.745619820771932, 8.538267681205997, 9.002944611147541, 9.621873520805828, 10.38148636378456, 10.95012506168618, 11.19743665250933, 11.85216296655959, 12.25799946675980, 12.78712154160166, 13.31033069627858, 14.18122894341237, 14.48214839860664, 15.21097736642868, 15.67553066579770
