| L(s) = 1 | + 3·3-s − 8·5-s + 9·9-s − 40·11-s − 4·13-s − 24·15-s + 84·17-s + 148·19-s − 84·23-s − 61·25-s + 27·27-s + 58·29-s − 136·31-s − 120·33-s − 222·37-s − 12·39-s − 420·41-s + 164·43-s − 72·45-s + 488·47-s + 252·51-s + 478·53-s + 320·55-s + 444·57-s + 548·59-s − 692·61-s + 32·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.715·5-s + 1/3·9-s − 1.09·11-s − 0.0853·13-s − 0.413·15-s + 1.19·17-s + 1.78·19-s − 0.761·23-s − 0.487·25-s + 0.192·27-s + 0.371·29-s − 0.787·31-s − 0.633·33-s − 0.986·37-s − 0.0492·39-s − 1.59·41-s + 0.581·43-s − 0.238·45-s + 1.51·47-s + 0.691·51-s + 1.23·53-s + 0.784·55-s + 1.03·57-s + 1.20·59-s − 1.45·61-s + 0.0610·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 148 T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 136 T + p^{3} T^{2} \) |
| 37 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 420 T + p^{3} T^{2} \) |
| 43 | \( 1 - 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 488 T + p^{3} T^{2} \) |
| 53 | \( 1 - 478 T + p^{3} T^{2} \) |
| 59 | \( 1 - 548 T + p^{3} T^{2} \) |
| 61 | \( 1 + 692 T + p^{3} T^{2} \) |
| 67 | \( 1 - 908 T + p^{3} T^{2} \) |
| 71 | \( 1 - 524 T + p^{3} T^{2} \) |
| 73 | \( 1 + 440 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1216 T + p^{3} T^{2} \) |
| 83 | \( 1 + 684 T + p^{3} T^{2} \) |
| 89 | \( 1 + 604 T + p^{3} T^{2} \) |
| 97 | \( 1 - 832 T + p^{3} T^{2} \) |
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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
−8.123645417557651195011298553208, −7.56182038391889833495723728413, −7.08629517177067581326398069215, −5.64303168233611032758649310676, −5.20447971059024151737074707558, −3.96670316688692151243223429294, −3.35086396408053148331437457934, −2.47012724712489243312849686543, −1.22647990755166036226127108245, 0,
1.22647990755166036226127108245, 2.47012724712489243312849686543, 3.35086396408053148331437457934, 3.96670316688692151243223429294, 5.20447971059024151737074707558, 5.64303168233611032758649310676, 7.08629517177067581326398069215, 7.56182038391889833495723728413, 8.123645417557651195011298553208
