| L(s) = 1 | + 4·4-s + 3·5-s − 7·7-s + 9·9-s + 13·13-s + 16·16-s − 25·19-s + 12·20-s − 45·23-s − 16·25-s − 28·28-s − 33·29-s + 55·31-s − 21·35-s + 36·36-s − 30·41-s − 5·43-s + 27·45-s − 81·47-s + 49·49-s + 52·52-s + 15·53-s + 90·59-s − 63·63-s + 64·64-s + 39·65-s − 29·73-s + ⋯ |
| L(s) = 1 | + 4-s + 3/5·5-s − 7-s + 9-s + 13-s + 16-s − 1.31·19-s + 3/5·20-s − 1.95·23-s − 0.639·25-s − 28-s − 1.13·29-s + 1.77·31-s − 3/5·35-s + 36-s − 0.731·41-s − 0.116·43-s + 3/5·45-s − 1.72·47-s + 49-s + 52-s + 0.283·53-s + 1.52·59-s − 63-s + 64-s + 3/5·65-s − 0.397·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.635130797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.635130797\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + p T \) |
| 13 | \( 1 - p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( 1 - 3 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 + 25 T + p^{2} T^{2} \) |
| 23 | \( 1 + 45 T + p^{2} T^{2} \) |
| 29 | \( 1 + 33 T + p^{2} T^{2} \) |
| 31 | \( 1 - 55 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( 1 + 30 T + p^{2} T^{2} \) |
| 43 | \( 1 + 5 T + p^{2} T^{2} \) |
| 47 | \( 1 + 81 T + p^{2} T^{2} \) |
| 53 | \( 1 - 15 T + p^{2} T^{2} \) |
| 59 | \( 1 - 90 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 29 T + p^{2} T^{2} \) |
| 79 | \( 1 - 67 T + p^{2} T^{2} \) |
| 83 | \( 1 - 159 T + p^{2} T^{2} \) |
| 89 | \( 1 + 165 T + p^{2} T^{2} \) |
| 97 | \( 1 - 131 T + p^{2} T^{2} \) |
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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
−13.63197053566900405627231663589, −12.84362764067401746008965065195, −11.75412343194729896810566923688, −10.39793021676748680743601017441, −9.802138136122261652972682765143, −8.138152755193870332660220056252, −6.65743424470249546613997148518, −6.04034891729060561075714472686, −3.80993439858702611792894369613, −1.98675976576929916867054592859,
1.98675976576929916867054592859, 3.80993439858702611792894369613, 6.04034891729060561075714472686, 6.65743424470249546613997148518, 8.138152755193870332660220056252, 9.802138136122261652972682765143, 10.39793021676748680743601017441, 11.75412343194729896810566923688, 12.84362764067401746008965065195, 13.63197053566900405627231663589
