L(s) = 1 | + 3·3-s + 4·5-s − 7·7-s + 9·9-s − 62·11-s + 62·13-s + 12·15-s + 84·17-s − 100·19-s − 21·21-s − 42·23-s − 109·25-s + 27·27-s + 10·29-s − 48·31-s − 186·33-s − 28·35-s + 246·37-s + 186·39-s − 248·41-s − 68·43-s + 36·45-s + 324·47-s + 49·49-s + 252·51-s − 258·53-s − 248·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.357·5-s − 0.377·7-s + 1/3·9-s − 1.69·11-s + 1.32·13-s + 0.206·15-s + 1.19·17-s − 1.20·19-s − 0.218·21-s − 0.380·23-s − 0.871·25-s + 0.192·27-s + 0.0640·29-s − 0.278·31-s − 0.981·33-s − 0.135·35-s + 1.09·37-s + 0.763·39-s − 0.944·41-s − 0.241·43-s + 0.119·45-s + 1.00·47-s + 1/7·49-s + 0.691·51-s − 0.668·53-s − 0.608·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
good | 5 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 62 T + p^{3} T^{2} \) |
| 13 | \( 1 - 62 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 42 T + p^{3} T^{2} \) |
| 29 | \( 1 - 10 T + p^{3} T^{2} \) |
| 31 | \( 1 + 48 T + p^{3} T^{2} \) |
| 37 | \( 1 - 246 T + p^{3} T^{2} \) |
| 41 | \( 1 + 248 T + p^{3} T^{2} \) |
| 43 | \( 1 + 68 T + p^{3} T^{2} \) |
| 47 | \( 1 - 324 T + p^{3} T^{2} \) |
| 53 | \( 1 + 258 T + p^{3} T^{2} \) |
| 59 | \( 1 + 120 T + p^{3} T^{2} \) |
| 61 | \( 1 + 622 T + p^{3} T^{2} \) |
| 67 | \( 1 + 904 T + p^{3} T^{2} \) |
| 71 | \( 1 + 678 T + p^{3} T^{2} \) |
| 73 | \( 1 + 642 T + p^{3} T^{2} \) |
| 79 | \( 1 - 740 T + p^{3} T^{2} \) |
| 83 | \( 1 + 468 T + p^{3} T^{2} \) |
| 89 | \( 1 - 200 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1266 T + p^{3} T^{2} \) |
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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
−8.739250510367619291138401834871, −8.063011484403483794150764682539, −7.43816577747085928414009935745, −6.16524378370530187799461487610, −5.68537232990306670572870835930, −4.47359154573672067135348034971, −3.44719888659052754622041816101, −2.61299624017185976784268651282, −1.52536861439271447949090332570, 0,
1.52536861439271447949090332570, 2.61299624017185976784268651282, 3.44719888659052754622041816101, 4.47359154573672067135348034971, 5.68537232990306670572870835930, 6.16524378370530187799461487610, 7.43816577747085928414009935745, 8.063011484403483794150764682539, 8.739250510367619291138401834871