| L(s) = 1 | + 3·3-s + 5·5-s − 7·7-s + 9·9-s + 36·11-s − 34·13-s + 15·15-s − 6·17-s + 28·19-s − 21·21-s − 192·23-s + 25·25-s + 27·27-s − 186·29-s − 176·31-s + 108·33-s − 35·35-s − 418·37-s − 102·39-s − 30·41-s + 412·43-s + 45·45-s + 432·47-s + 49·49-s − 18·51-s − 306·53-s + 180·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.986·11-s − 0.725·13-s + 0.258·15-s − 0.0856·17-s + 0.338·19-s − 0.218·21-s − 1.74·23-s + 1/5·25-s + 0.192·27-s − 1.19·29-s − 1.01·31-s + 0.569·33-s − 0.169·35-s − 1.85·37-s − 0.418·39-s − 0.114·41-s + 1.46·43-s + 0.149·45-s + 1.34·47-s + 1/7·49-s − 0.0494·51-s − 0.793·53-s + 0.441·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 - 28 T + p^{3} T^{2} \) |
| 23 | \( 1 + 192 T + p^{3} T^{2} \) |
| 29 | \( 1 + 186 T + p^{3} T^{2} \) |
| 31 | \( 1 + 176 T + p^{3} T^{2} \) |
| 37 | \( 1 + 418 T + p^{3} T^{2} \) |
| 41 | \( 1 + 30 T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 432 T + p^{3} T^{2} \) |
| 53 | \( 1 + 306 T + p^{3} T^{2} \) |
| 59 | \( 1 - 564 T + p^{3} T^{2} \) |
| 61 | \( 1 + 322 T + p^{3} T^{2} \) |
| 67 | \( 1 + 716 T + p^{3} T^{2} \) |
| 71 | \( 1 - 48 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1078 T + p^{3} T^{2} \) |
| 79 | \( 1 - 496 T + p^{3} T^{2} \) |
| 83 | \( 1 + 468 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1314 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1438 T + p^{3} T^{2} \) |
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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
−8.877419756264114280107447240699, −7.68402058540767292432463299405, −7.13358370289385332060984709067, −6.16276256619934687169032301559, −5.42112933145253168721403330339, −4.19819467750920188469456903929, −3.53076618434792396319063927122, −2.35957722486011567083083883592, −1.54436099878360210227739416771, 0,
1.54436099878360210227739416771, 2.35957722486011567083083883592, 3.53076618434792396319063927122, 4.19819467750920188469456903929, 5.42112933145253168721403330339, 6.16276256619934687169032301559, 7.13358370289385332060984709067, 7.68402058540767292432463299405, 8.877419756264114280107447240699
