| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 11-s − 2·13-s + 16-s − 17-s − 4·19-s + 2·20-s − 22-s − 25-s + 2·26-s + 2·29-s − 8·31-s − 32-s + 34-s − 10·37-s + 4·38-s − 2·40-s + 6·41-s + 4·43-s + 44-s − 8·47-s − 7·49-s + 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 0.301·11-s − 0.554·13-s + 1/4·16-s − 0.242·17-s − 0.917·19-s + 0.447·20-s − 0.213·22-s − 1/5·25-s + 0.392·26-s + 0.371·29-s − 1.43·31-s − 0.176·32-s + 0.171·34-s − 1.64·37-s + 0.648·38-s − 0.316·40-s + 0.937·41-s + 0.609·43-s + 0.150·44-s − 1.16·47-s − 49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.399296447116303767909000407556, −7.51069360070726706710817030189, −6.79674128443302554914106096661, −6.11520564042065569362180516338, −5.37583212972271956479917969694, −4.41244970448294646210518381879, −3.30186553022733953754225590472, −2.22811621544823531394293472718, −1.58149005341881866981691867207, 0,
1.58149005341881866981691867207, 2.22811621544823531394293472718, 3.30186553022733953754225590472, 4.41244970448294646210518381879, 5.37583212972271956479917969694, 6.11520564042065569362180516338, 6.79674128443302554914106096661, 7.51069360070726706710817030189, 8.399296447116303767909000407556
