L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 6·11-s + 13-s − 15-s − 6·17-s − 7·19-s − 21-s − 6·23-s + 25-s − 27-s − 9·29-s − 4·31-s + 6·33-s + 35-s − 39-s + 6·41-s + 2·43-s + 45-s − 12·47-s − 6·49-s + 6·51-s + 6·53-s − 6·55-s + 7·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 0.277·13-s − 0.258·15-s − 1.45·17-s − 1.60·19-s − 0.218·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.67·29-s − 0.718·31-s + 1.04·33-s + 0.169·35-s − 0.160·39-s + 0.937·41-s + 0.304·43-s + 0.149·45-s − 1.75·47-s − 6/7·49-s + 0.840·51-s + 0.824·53-s − 0.809·55-s + 0.927·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.95750991922508, −12.52979607755882, −11.90137293121833, −11.22539623030694, −10.99294482365790, −10.66941332802551, −10.30872063441006, −9.587294074844549, −9.332674252576116, −8.578720198404945, −8.271545368979424, −7.729223920078316, −7.357833172344576, −6.659509579220553, −6.174809863365255, −5.908378695707729, −5.243283670525342, −4.913634957848484, −4.327628697650604, −3.917032559083439, −3.155549360996183, −2.382840304701651, −2.000718299097410, −1.708665528473256, −0.4616609075085410, 0,
0.4616609075085410, 1.708665528473256, 2.000718299097410, 2.382840304701651, 3.155549360996183, 3.917032559083439, 4.327628697650604, 4.913634957848484, 5.243283670525342, 5.908378695707729, 6.174809863365255, 6.659509579220553, 7.357833172344576, 7.729223920078316, 8.271545368979424, 8.578720198404945, 9.332674252576116, 9.587294074844549, 10.30872063441006, 10.66941332802551, 10.99294482365790, 11.22539623030694, 11.90137293121833, 12.52979607755882, 12.95750991922508