| L(s) = 1 | + 3-s − 2·4-s + 9-s − 2·12-s − 13-s + 4·16-s + 6·17-s − 5·19-s − 6·23-s + 27-s − 6·29-s − 5·31-s − 2·36-s + 7·37-s − 39-s − 12·41-s + 43-s + 6·47-s + 4·48-s + 6·51-s + 2·52-s − 5·57-s + 6·59-s − 2·61-s − 8·64-s + 7·67-s − 12·68-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1/3·9-s − 0.577·12-s − 0.277·13-s + 16-s + 1.45·17-s − 1.14·19-s − 1.25·23-s + 0.192·27-s − 1.11·29-s − 0.898·31-s − 1/3·36-s + 1.15·37-s − 0.160·39-s − 1.87·41-s + 0.152·43-s + 0.875·47-s + 0.577·48-s + 0.840·51-s + 0.277·52-s − 0.662·57-s + 0.781·59-s − 0.256·61-s − 64-s + 0.855·67-s − 1.45·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 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
−8.135245437116732921186666216165, −7.75110803385765164493233092171, −6.75870879468165706986788693687, −5.71712970663256425424326189478, −5.17862009613309092684526088674, −4.02346664539105473677147683060, −3.74662313532088859915068806107, −2.55130249126529936360206208492, −1.44711402334891860578104024553, 0,
1.44711402334891860578104024553, 2.55130249126529936360206208492, 3.74662313532088859915068806107, 4.02346664539105473677147683060, 5.17862009613309092684526088674, 5.71712970663256425424326189478, 6.75870879468165706986788693687, 7.75110803385765164493233092171, 8.135245437116732921186666216165
