| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s − 13-s + 2·15-s − 4·17-s + 8·19-s − 21-s − 6·23-s − 25-s − 27-s + 2·29-s + 8·31-s − 2·35-s − 2·37-s + 39-s − 2·41-s + 4·43-s − 2·45-s − 2·47-s + 49-s + 4·51-s − 6·53-s − 8·57-s + 6·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.516·15-s − 0.970·17-s + 1.83·19-s − 0.218·21-s − 1.25·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.338·35-s − 0.328·37-s + 0.160·39-s − 0.312·41-s + 0.609·43-s − 0.298·45-s − 0.291·47-s + 1/7·49-s + 0.560·51-s − 0.824·53-s − 1.05·57-s + 0.781·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.936543601978263475831382319040, −7.34107616517426769606325482661, −6.59150755013839775303486722074, −5.76832013105942246038988781129, −4.93892093290200253084127665689, −4.32290886022282521215654462379, −3.51779498018495891444818356989, −2.44479448891019510347240980858, −1.20600151254866230461336296054, 0,
1.20600151254866230461336296054, 2.44479448891019510347240980858, 3.51779498018495891444818356989, 4.32290886022282521215654462379, 4.93892093290200253084127665689, 5.76832013105942246038988781129, 6.59150755013839775303486722074, 7.34107616517426769606325482661, 7.936543601978263475831382319040
