L(s) = 1 | − 8·4-s − 12·5-s − 2·7-s − 36·11-s + 64·16-s + 78·17-s − 74·19-s + 96·20-s + 96·23-s + 19·25-s + 16·28-s − 18·29-s + 214·31-s + 24·35-s + 286·37-s − 384·41-s + 524·43-s + 288·44-s + 300·47-s − 339·49-s − 558·53-s + 432·55-s + 576·59-s + 74·61-s − 512·64-s − 38·67-s − 624·68-s + ⋯ |
L(s) = 1 | − 4-s − 1.07·5-s − 0.107·7-s − 0.986·11-s + 16-s + 1.11·17-s − 0.893·19-s + 1.07·20-s + 0.870·23-s + 0.151·25-s + 0.107·28-s − 0.115·29-s + 1.23·31-s + 0.115·35-s + 1.27·37-s − 1.46·41-s + 1.85·43-s + 0.986·44-s + 0.931·47-s − 0.988·49-s − 1.44·53-s + 1.05·55-s + 1.27·59-s + 0.155·61-s − 64-s − 0.0692·67-s − 1.11·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + p^{3} T^{2} \) |
| 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 17 | \( 1 - 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 74 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 18 T + p^{3} T^{2} \) |
| 31 | \( 1 - 214 T + p^{3} T^{2} \) |
| 37 | \( 1 - 286 T + p^{3} T^{2} \) |
| 41 | \( 1 + 384 T + p^{3} T^{2} \) |
| 43 | \( 1 - 524 T + p^{3} T^{2} \) |
| 47 | \( 1 - 300 T + p^{3} T^{2} \) |
| 53 | \( 1 + 558 T + p^{3} T^{2} \) |
| 59 | \( 1 - 576 T + p^{3} T^{2} \) |
| 61 | \( 1 - 74 T + p^{3} T^{2} \) |
| 67 | \( 1 + 38 T + p^{3} T^{2} \) |
| 71 | \( 1 + 456 T + p^{3} T^{2} \) |
| 73 | \( 1 - 682 T + p^{3} T^{2} \) |
| 79 | \( 1 - 704 T + p^{3} T^{2} \) |
| 83 | \( 1 + 888 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1020 T + p^{3} T^{2} \) |
| 97 | \( 1 + 110 T + p^{3} T^{2} \) |
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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
−8.506849628845344723514082586627, −8.011639279915489978014522100889, −7.38653181430732472642448446296, −6.15555649546157218023010101946, −5.19323643456522147520753545141, −4.46641162489254132942475963124, −3.65279988902516954528815536248, −2.73077219332196411843707325861, −0.957253493927889040989388724980, 0,
0.957253493927889040989388724980, 2.73077219332196411843707325861, 3.65279988902516954528815536248, 4.46641162489254132942475963124, 5.19323643456522147520753545141, 6.15555649546157218023010101946, 7.38653181430732472642448446296, 8.011639279915489978014522100889, 8.506849628845344723514082586627