L(s) = 1 | + 3-s + 4·7-s − 2·9-s − 3·11-s + 17-s − 7·19-s + 4·21-s + 4·23-s − 5·27-s − 8·29-s − 4·31-s − 3·33-s + 4·37-s − 3·41-s + 8·43-s + 9·49-s + 51-s − 12·53-s − 7·57-s − 8·59-s + 4·61-s − 8·63-s − 9·67-s + 4·69-s − 16·71-s + 11·73-s − 12·77-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.51·7-s − 2/3·9-s − 0.904·11-s + 0.242·17-s − 1.60·19-s + 0.872·21-s + 0.834·23-s − 0.962·27-s − 1.48·29-s − 0.718·31-s − 0.522·33-s + 0.657·37-s − 0.468·41-s + 1.21·43-s + 9/7·49-s + 0.140·51-s − 1.64·53-s − 0.927·57-s − 1.04·59-s + 0.512·61-s − 1.00·63-s − 1.09·67-s + 0.481·69-s − 1.89·71-s + 1.28·73-s − 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.78615886212200770984375397490, −7.29374988162258519896047345493, −6.11370740469943774195157524925, −5.49001155550984450986591493691, −4.78573682531592981607872028344, −4.07856995748849918073068101877, −3.06168367496202186229726237042, −2.27768286996207507182177951187, −1.57342011591946004612772009438, 0,
1.57342011591946004612772009438, 2.27768286996207507182177951187, 3.06168367496202186229726237042, 4.07856995748849918073068101877, 4.78573682531592981607872028344, 5.49001155550984450986591493691, 6.11370740469943774195157524925, 7.29374988162258519896047345493, 7.78615886212200770984375397490