L(s) = 1 | + 7-s − 3·9-s + 4·11-s + 3·13-s − 17-s + 23-s + 4·31-s − 11·37-s − 10·41-s − 2·43-s − 11·47-s + 49-s − 53-s − 8·59-s + 8·61-s − 3·63-s + 4·71-s − 4·73-s + 4·77-s + 11·79-s + 9·81-s − 13·83-s + 89-s + 3·91-s − 7·97-s − 12·99-s + 101-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 9-s + 1.20·11-s + 0.832·13-s − 0.242·17-s + 0.208·23-s + 0.718·31-s − 1.80·37-s − 1.56·41-s − 0.304·43-s − 1.60·47-s + 1/7·49-s − 0.137·53-s − 1.04·59-s + 1.02·61-s − 0.377·63-s + 0.474·71-s − 0.468·73-s + 0.455·77-s + 1.23·79-s + 81-s − 1.42·83-s + 0.105·89-s + 0.314·91-s − 0.710·97-s − 1.20·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.13361089625145, −12.50356326969170, −11.98706671498460, −11.65235545320724, −11.31599799574421, −10.85692331674081, −10.31731903261871, −9.811292261616964, −9.255565406910189, −8.755153402460350, −8.407958585839633, −8.188956771334209, −7.347617557537979, −6.798505988449944, −6.443636187138401, −6.024916529286399, −5.337726739470262, −4.959750802279539, −4.384893703737495, −3.641824267073030, −3.389903646314907, −2.801192298349212, −1.860447455229482, −1.609983513902030, −0.8039444628185411, 0,
0.8039444628185411, 1.609983513902030, 1.860447455229482, 2.801192298349212, 3.389903646314907, 3.641824267073030, 4.384893703737495, 4.959750802279539, 5.337726739470262, 6.024916529286399, 6.443636187138401, 6.798505988449944, 7.347617557537979, 8.188956771334209, 8.407958585839633, 8.755153402460350, 9.255565406910189, 9.811292261616964, 10.31731903261871, 10.85692331674081, 11.31599799574421, 11.65235545320724, 11.98706671498460, 12.50356326969170, 13.13361089625145