| L(s) = 1 | + 3·3-s + 27.1·7-s + 9·9-s − 32.2·11-s − 42.0·13-s − 88.5·17-s + 36.6·19-s + 81.4·21-s + 140.·23-s + 27·27-s − 164.·29-s − 80.7·31-s − 96.6·33-s − 126.·37-s − 126.·39-s + 285.·41-s + 249.·43-s − 481.·47-s + 394.·49-s − 265.·51-s − 560.·53-s + 109.·57-s + 308.·59-s − 680.·61-s + 244.·63-s − 501.·67-s + 421.·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.46·7-s + 0.333·9-s − 0.883·11-s − 0.897·13-s − 1.26·17-s + 0.442·19-s + 0.846·21-s + 1.27·23-s + 0.192·27-s − 1.05·29-s − 0.468·31-s − 0.510·33-s − 0.562·37-s − 0.518·39-s + 1.08·41-s + 0.884·43-s − 1.49·47-s + 1.15·49-s − 0.729·51-s − 1.45·53-s + 0.255·57-s + 0.680·59-s − 1.42·61-s + 0.488·63-s − 0.913·67-s + 0.736·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 27.1T + 343T^{2} \) |
| 11 | \( 1 + 32.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 42.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 88.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 36.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 164.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 80.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 126.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 285.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 249.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 481.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 560.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 308.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 680.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 501.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 259.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 732.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 855.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 593.T + 9.12e5T^{2} \) |
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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
−8.149469059498190078517818316611, −7.55533859446765468263116308515, −7.01417373536375801412945681792, −5.70020974930905767099539156838, −4.87224955342230388488983979998, −4.43894718464775464179746511090, −3.11841987204721524222948248240, −2.27407903275604145623744135236, −1.46778216758869320345702824695, 0,
1.46778216758869320345702824695, 2.27407903275604145623744135236, 3.11841987204721524222948248240, 4.43894718464775464179746511090, 4.87224955342230388488983979998, 5.70020974930905767099539156838, 7.01417373536375801412945681792, 7.55533859446765468263116308515, 8.149469059498190078517818316611
