| L(s) = 1 | − 3-s + 7-s − 2·9-s − 3·11-s + 4·13-s − 4·19-s − 21-s + 5·27-s + 2·31-s + 3·33-s − 37-s − 4·39-s + 3·41-s − 2·43-s − 3·47-s − 6·49-s + 9·53-s + 4·57-s + 2·61-s − 2·63-s + 4·67-s + 15·71-s + 7·73-s − 3·77-s − 10·79-s + 81-s + 3·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s − 0.904·11-s + 1.10·13-s − 0.917·19-s − 0.218·21-s + 0.962·27-s + 0.359·31-s + 0.522·33-s − 0.164·37-s − 0.640·39-s + 0.468·41-s − 0.304·43-s − 0.437·47-s − 6/7·49-s + 1.23·53-s + 0.529·57-s + 0.256·61-s − 0.251·63-s + 0.488·67-s + 1.78·71-s + 0.819·73-s − 0.341·77-s − 1.12·79-s + 1/9·81-s + 0.329·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223425879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223425879\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.320566320344959555589170558707, −8.063386549062472212301129813398, −6.88610542134705476043905926803, −6.23303933687777995129974330697, −5.53161304223054046550311903611, −4.89610125391740010196132513501, −3.94769425088528710321117034658, −2.96736714350579454573712100915, −1.98523078966066915455111948391, −0.65226138130976440323790882395,
0.65226138130976440323790882395, 1.98523078966066915455111948391, 2.96736714350579454573712100915, 3.94769425088528710321117034658, 4.89610125391740010196132513501, 5.53161304223054046550311903611, 6.23303933687777995129974330697, 6.88610542134705476043905926803, 8.063386549062472212301129813398, 8.320566320344959555589170558707
