| L(s) = 1 | − 3·3-s − 8·7-s + 9·9-s + 36·11-s + 10·13-s − 18·17-s − 100·19-s + 24·21-s − 72·23-s − 27·27-s − 234·29-s − 16·31-s − 108·33-s + 226·37-s − 30·39-s + 90·41-s − 452·43-s − 432·47-s − 279·49-s + 54·51-s − 414·53-s + 300·57-s − 684·59-s + 422·61-s − 72·63-s − 332·67-s + 216·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.431·7-s + 1/3·9-s + 0.986·11-s + 0.213·13-s − 0.256·17-s − 1.20·19-s + 0.249·21-s − 0.652·23-s − 0.192·27-s − 1.49·29-s − 0.0926·31-s − 0.569·33-s + 1.00·37-s − 0.123·39-s + 0.342·41-s − 1.60·43-s − 1.34·47-s − 0.813·49-s + 0.148·51-s − 1.07·53-s + 0.697·57-s − 1.50·59-s + 0.885·61-s − 0.143·63-s − 0.605·67-s + 0.376·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 234 T + p^{3} T^{2} \) |
| 31 | \( 1 + 16 T + p^{3} T^{2} \) |
| 37 | \( 1 - 226 T + p^{3} T^{2} \) |
| 41 | \( 1 - 90 T + p^{3} T^{2} \) |
| 43 | \( 1 + 452 T + p^{3} T^{2} \) |
| 47 | \( 1 + 432 T + p^{3} T^{2} \) |
| 53 | \( 1 + 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 684 T + p^{3} T^{2} \) |
| 61 | \( 1 - 422 T + p^{3} T^{2} \) |
| 67 | \( 1 + 332 T + p^{3} T^{2} \) |
| 71 | \( 1 + 360 T + p^{3} T^{2} \) |
| 73 | \( 1 + 26 T + p^{3} T^{2} \) |
| 79 | \( 1 - 512 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 630 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1054 T + p^{3} T^{2} \) |
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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
−10.98683723911199530993692302620, −9.899044986010273771857459582515, −9.084903811470762423135826182892, −7.914711183112606979421213940628, −6.63000214420936177439038483043, −6.06640516989865922907789152481, −4.64739796747380623099202913633, −3.57115521583206568695402412634, −1.75713841966692905912181286596, 0,
1.75713841966692905912181286596, 3.57115521583206568695402412634, 4.64739796747380623099202913633, 6.06640516989865922907789152481, 6.63000214420936177439038483043, 7.914711183112606979421213940628, 9.084903811470762423135826182892, 9.899044986010273771857459582515, 10.98683723911199530993692302620
