L(s) = 1 | − 2·5-s − 7-s + 4·11-s − 6·13-s − 2·17-s − 19-s + 4·23-s − 25-s − 6·29-s + 4·31-s + 2·35-s + 2·37-s − 10·41-s + 12·43-s − 4·47-s + 49-s − 14·53-s − 8·55-s + 4·59-s + 6·61-s + 12·65-s + 4·67-s + 8·71-s − 6·73-s − 4·77-s − 12·79-s − 12·83-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.377·7-s + 1.20·11-s − 1.66·13-s − 0.485·17-s − 0.229·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.338·35-s + 0.328·37-s − 1.56·41-s + 1.82·43-s − 0.583·47-s + 1/7·49-s − 1.92·53-s − 1.07·55-s + 0.520·59-s + 0.768·61-s + 1.48·65-s + 0.488·67-s + 0.949·71-s − 0.702·73-s − 0.455·77-s − 1.35·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9976456837\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9976456837\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.56104307847507914306941164567, −7.04963412034959358049338977107, −6.52721406711519652070455212212, −5.63463860576601220219622348360, −4.74315867444121788194074534907, −4.23452993120533202737487806333, −3.49127575348744521609247307953, −2.69769996554166863344907061341, −1.73447558522398710371506885094, −0.46654872348171173356770917177,
0.46654872348171173356770917177, 1.73447558522398710371506885094, 2.69769996554166863344907061341, 3.49127575348744521609247307953, 4.23452993120533202737487806333, 4.74315867444121788194074534907, 5.63463860576601220219622348360, 6.52721406711519652070455212212, 7.04963412034959358049338977107, 7.56104307847507914306941164567