| L(s) = 1 | + 5-s + 2·7-s − 3·11-s + 3·13-s − 17-s + 2·19-s + 23-s + 25-s − 9·31-s + 2·35-s − 12·37-s − 41-s − 43-s + 8·47-s − 3·49-s + 5·53-s − 3·55-s + 12·59-s − 12·61-s + 3·65-s + 13·67-s − 6·71-s − 8·73-s − 6·77-s + 8·79-s − 3·83-s − 85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.904·11-s + 0.832·13-s − 0.242·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s − 1.61·31-s + 0.338·35-s − 1.97·37-s − 0.156·41-s − 0.152·43-s + 1.16·47-s − 3/7·49-s + 0.686·53-s − 0.404·55-s + 1.56·59-s − 1.53·61-s + 0.372·65-s + 1.58·67-s − 0.712·71-s − 0.936·73-s − 0.683·77-s + 0.900·79-s − 0.329·83-s − 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.40508599584318, −14.74775773723847, −14.35328278538031, −13.62039793711404, −13.42980242689407, −12.78792806327736, −12.17538629988886, −11.62774462726336, −10.87713426474963, −10.72287003471362, −10.11028038598013, −9.341532753424616, −8.832313303479611, −8.361111931766132, −7.718410981682325, −7.147468695968244, −6.593575473305274, −5.670916338352502, −5.389623447823603, −4.838329465857430, −3.938972486720594, −3.388702037377199, −2.509575213961978, −1.848048259882298, −1.170575475517847, 0,
1.170575475517847, 1.848048259882298, 2.509575213961978, 3.388702037377199, 3.938972486720594, 4.838329465857430, 5.389623447823603, 5.670916338352502, 6.593575473305274, 7.147468695968244, 7.718410981682325, 8.361111931766132, 8.832313303479611, 9.341532753424616, 10.11028038598013, 10.72287003471362, 10.87713426474963, 11.62774462726336, 12.17538629988886, 12.78792806327736, 13.42980242689407, 13.62039793711404, 14.35328278538031, 14.74775773723847, 15.40508599584318
