L(s) = 1 | + 2·3-s − 9·5-s + 31·7-s − 23·9-s − 57·11-s − 52·13-s − 18·15-s + 69·17-s − 19·19-s + 62·21-s + 72·23-s − 44·25-s − 100·27-s − 150·29-s − 32·31-s − 114·33-s − 279·35-s − 226·37-s − 104·39-s − 258·41-s + 67·43-s + 207·45-s − 579·47-s + 618·49-s + 138·51-s − 432·53-s + 513·55-s + ⋯ |
L(s) = 1 | + 0.384·3-s − 0.804·5-s + 1.67·7-s − 0.851·9-s − 1.56·11-s − 1.10·13-s − 0.309·15-s + 0.984·17-s − 0.229·19-s + 0.644·21-s + 0.652·23-s − 0.351·25-s − 0.712·27-s − 0.960·29-s − 0.185·31-s − 0.601·33-s − 1.34·35-s − 1.00·37-s − 0.427·39-s − 0.982·41-s + 0.237·43-s + 0.685·45-s − 1.79·47-s + 1.80·49-s + 0.378·51-s − 1.11·53-s + 1.25·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + p T \) |
good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 + 9 T + p^{3} T^{2} \) |
| 7 | \( 1 - 31 T + p^{3} T^{2} \) |
| 11 | \( 1 + 57 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 69 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 150 T + p^{3} T^{2} \) |
| 31 | \( 1 + 32 T + p^{3} T^{2} \) |
| 37 | \( 1 + 226 T + p^{3} T^{2} \) |
| 41 | \( 1 + 258 T + p^{3} T^{2} \) |
| 43 | \( 1 - 67 T + p^{3} T^{2} \) |
| 47 | \( 1 + 579 T + p^{3} T^{2} \) |
| 53 | \( 1 + 432 T + p^{3} T^{2} \) |
| 59 | \( 1 - 330 T + p^{3} T^{2} \) |
| 61 | \( 1 + 13 T + p^{3} T^{2} \) |
| 67 | \( 1 - 856 T + p^{3} T^{2} \) |
| 71 | \( 1 + 642 T + p^{3} T^{2} \) |
| 73 | \( 1 + 487 T + p^{3} T^{2} \) |
| 79 | \( 1 - 700 T + p^{3} T^{2} \) |
| 83 | \( 1 - 12 T + p^{3} T^{2} \) |
| 89 | \( 1 + 600 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1424 T + p^{3} T^{2} \) |
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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
−11.02765498720393380977754845691, −9.958849104015587500455733001255, −8.562369312881309059590822425295, −7.922508405844637278995146756192, −7.42603270077594087915398038312, −5.41461013002335235077885845852, −4.85691004697971132616812533891, −3.30444151547085117275440013323, −2.04294387833356530883690408960, 0,
2.04294387833356530883690408960, 3.30444151547085117275440013323, 4.85691004697971132616812533891, 5.41461013002335235077885845852, 7.42603270077594087915398038312, 7.922508405844637278995146756192, 8.562369312881309059590822425295, 9.958849104015587500455733001255, 11.02765498720393380977754845691