L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 2·13-s + 15-s − 2·17-s + 6·19-s − 21-s + 25-s + 27-s − 2·29-s + 4·31-s − 35-s + 12·37-s + 2·39-s − 10·41-s + 12·43-s + 45-s + 4·47-s + 49-s − 2·51-s + 2·53-s + 6·57-s − 6·59-s + 2·61-s − 63-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s − 0.485·17-s + 1.37·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.169·35-s + 1.97·37-s + 0.320·39-s − 1.56·41-s + 1.82·43-s + 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.274·53-s + 0.794·57-s − 0.781·59-s + 0.256·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.39185003442604, −12.88821293156223, −12.29729891019482, −11.79448466792817, −11.39492571882904, −10.73310491622837, −10.43971037595065, −9.694030183892480, −9.518585863062845, −9.051238479209323, −8.557718459252198, −7.954662849702894, −7.579479759838017, −7.041767324140445, −6.448472938331811, −6.085065350767922, −5.462228535442207, −5.025521164229906, −4.217796789950285, −3.930556268791045, −3.211173781990770, −2.654367109839020, −2.355989953349923, −1.306409914012896, −1.092534382445246, 0,
1.092534382445246, 1.306409914012896, 2.355989953349923, 2.654367109839020, 3.211173781990770, 3.930556268791045, 4.217796789950285, 5.025521164229906, 5.462228535442207, 6.085065350767922, 6.448472938331811, 7.041767324140445, 7.579479759838017, 7.954662849702894, 8.557718459252198, 9.051238479209323, 9.518585863062845, 9.694030183892480, 10.43971037595065, 10.73310491622837, 11.39492571882904, 11.79448466792817, 12.29729891019482, 12.88821293156223, 13.39185003442604