| L(s) = 1 | − 3-s − 4·7-s + 9-s + 4·13-s + 6·17-s − 19-s + 4·21-s − 6·23-s − 5·25-s − 27-s − 6·29-s + 2·31-s + 4·37-s − 4·39-s + 6·41-s + 4·43-s + 6·47-s + 9·49-s − 6·51-s − 6·53-s + 57-s + 12·59-s − 14·61-s − 4·63-s − 8·67-s + 6·69-s + 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.10·13-s + 1.45·17-s − 0.229·19-s + 0.872·21-s − 1.25·23-s − 25-s − 0.192·27-s − 1.11·29-s + 0.359·31-s + 0.657·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s + 0.875·47-s + 9/7·49-s − 0.840·51-s − 0.824·53-s + 0.132·57-s + 1.56·59-s − 1.79·61-s − 0.503·63-s − 0.977·67-s + 0.722·69-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.992680376128817439410616241148, −7.47620423165504362047840493205, −6.37837332665094581753674197922, −6.03723114482210006778449723144, −5.45803980005957483798159176539, −4.03060027612472363140407850727, −3.67844472597115610426331813053, −2.59451510562992864284961947324, −1.24718934600363714757600412295, 0,
1.24718934600363714757600412295, 2.59451510562992864284961947324, 3.67844472597115610426331813053, 4.03060027612472363140407850727, 5.45803980005957483798159176539, 6.03723114482210006778449723144, 6.37837332665094581753674197922, 7.47620423165504362047840493205, 7.992680376128817439410616241148
