L(s) = 1 | − 3·7-s + 5·13-s − 5·19-s − 4·23-s − 4·29-s + 5·31-s + 10·37-s + 10·41-s + 43-s + 2·47-s + 2·49-s − 10·53-s − 10·59-s − 5·61-s − 3·67-s − 10·71-s + 10·73-s + 14·83-s − 16·89-s − 15·91-s + 5·97-s − 2·101-s + 16·103-s − 18·107-s + 5·109-s − 20·113-s + ⋯ |
L(s) = 1 | − 1.13·7-s + 1.38·13-s − 1.14·19-s − 0.834·23-s − 0.742·29-s + 0.898·31-s + 1.64·37-s + 1.56·41-s + 0.152·43-s + 0.291·47-s + 2/7·49-s − 1.37·53-s − 1.30·59-s − 0.640·61-s − 0.366·67-s − 1.18·71-s + 1.17·73-s + 1.53·83-s − 1.69·89-s − 1.57·91-s + 0.507·97-s − 0.199·101-s + 1.57·103-s − 1.74·107-s + 0.478·109-s − 1.88·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−7.74511570905981927975242879944, −6.60980982308190447217561867925, −6.20477719650650916055255171140, −5.79628178444275253255816919845, −4.50482885971862905864342104608, −3.97506824003525535463728850051, −3.17427009240857360583909442557, −2.36580254103133799659549033887, −1.21951238561099457541708383809, 0,
1.21951238561099457541708383809, 2.36580254103133799659549033887, 3.17427009240857360583909442557, 3.97506824003525535463728850051, 4.50482885971862905864342104608, 5.79628178444275253255816919845, 6.20477719650650916055255171140, 6.60980982308190447217561867925, 7.74511570905981927975242879944