L(s) = 1 | − 3-s + 3·7-s − 2·9-s + 2·11-s − 13-s − 5·17-s − 3·21-s − 23-s − 5·25-s + 5·27-s + 3·29-s − 4·31-s − 2·33-s − 2·37-s + 39-s + 8·41-s − 8·43-s − 8·47-s + 2·49-s + 5·51-s − 9·53-s − 59-s + 14·61-s − 6·63-s − 13·67-s + 69-s − 10·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.13·7-s − 2/3·9-s + 0.603·11-s − 0.277·13-s − 1.21·17-s − 0.654·21-s − 0.208·23-s − 25-s + 0.962·27-s + 0.557·29-s − 0.718·31-s − 0.348·33-s − 0.328·37-s + 0.160·39-s + 1.24·41-s − 1.21·43-s − 1.16·47-s + 2/7·49-s + 0.700·51-s − 1.23·53-s − 0.130·59-s + 1.79·61-s − 0.755·63-s − 1.58·67-s + 0.120·69-s − 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.359566813592776605697943643675, −7.74613152589584229848027600152, −6.72598040565036698033673525843, −6.13690005259013972771512187418, −5.20164006899702178333342847264, −4.65444054474886627728449087886, −3.71573461775705014184176995236, −2.44767093282979086637766710358, −1.50567723507471670967120404283, 0,
1.50567723507471670967120404283, 2.44767093282979086637766710358, 3.71573461775705014184176995236, 4.65444054474886627728449087886, 5.20164006899702178333342847264, 6.13690005259013972771512187418, 6.72598040565036698033673525843, 7.74613152589584229848027600152, 8.359566813592776605697943643675