| L(s) = 1 | + 3·3-s − 18·5-s − 7·7-s + 9·9-s + 36·11-s − 34·13-s − 54·15-s + 42·17-s + 124·19-s − 21·21-s + 199·25-s + 27·27-s + 102·29-s + 160·31-s + 108·33-s + 126·35-s + 398·37-s − 102·39-s − 318·41-s + 268·43-s − 162·45-s − 240·47-s + 49·49-s + 126·51-s − 498·53-s − 648·55-s + 372·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.60·5-s − 0.377·7-s + 1/3·9-s + 0.986·11-s − 0.725·13-s − 0.929·15-s + 0.599·17-s + 1.49·19-s − 0.218·21-s + 1.59·25-s + 0.192·27-s + 0.653·29-s + 0.926·31-s + 0.569·33-s + 0.608·35-s + 1.76·37-s − 0.418·39-s − 1.21·41-s + 0.950·43-s − 0.536·45-s − 0.744·47-s + 1/7·49-s + 0.345·51-s − 1.29·53-s − 1.58·55-s + 0.864·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.676142833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.676142833\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 102 T + p^{3} T^{2} \) |
| 31 | \( 1 - 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 398 T + p^{3} T^{2} \) |
| 41 | \( 1 + 318 T + p^{3} T^{2} \) |
| 43 | \( 1 - 268 T + p^{3} T^{2} \) |
| 47 | \( 1 + 240 T + p^{3} T^{2} \) |
| 53 | \( 1 + 498 T + p^{3} T^{2} \) |
| 59 | \( 1 - 132 T + p^{3} T^{2} \) |
| 61 | \( 1 - 398 T + p^{3} T^{2} \) |
| 67 | \( 1 + 92 T + p^{3} T^{2} \) |
| 71 | \( 1 - 720 T + p^{3} T^{2} \) |
| 73 | \( 1 + 502 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1024 T + p^{3} T^{2} \) |
| 83 | \( 1 - 204 T + p^{3} T^{2} \) |
| 89 | \( 1 - 354 T + p^{3} T^{2} \) |
| 97 | \( 1 + 286 T + p^{3} T^{2} \) |
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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
−11.43808645846862009442108850231, −10.02564470085939608724930382948, −9.253989239769519311151869418084, −8.108753058559954913654317784987, −7.52899283861443377884794982206, −6.54325801361683782115600262154, −4.84914072840988045227180385771, −3.81155493369603893403057077384, −2.96531272172107702079338172136, −0.877631644606296641782168486277,
0.877631644606296641782168486277, 2.96531272172107702079338172136, 3.81155493369603893403057077384, 4.84914072840988045227180385771, 6.54325801361683782115600262154, 7.52899283861443377884794982206, 8.108753058559954913654317784987, 9.253989239769519311151869418084, 10.02564470085939608724930382948, 11.43808645846862009442108850231
