| L(s) = 1 | − 3-s + 9-s − 13-s − 17-s − 27-s − 2·29-s − 8·31-s − 10·37-s + 39-s − 6·41-s − 4·43-s − 8·47-s − 7·49-s + 51-s + 2·53-s + 4·59-s − 10·61-s + 8·67-s + 12·71-s − 6·73-s + 8·79-s + 81-s + 12·83-s + 2·87-s − 10·89-s + 8·93-s − 6·97-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.277·13-s − 0.242·17-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 1.64·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s − 1.16·47-s − 49-s + 0.140·51-s + 0.274·53-s + 0.520·59-s − 1.28·61-s + 0.977·67-s + 1.42·71-s − 0.702·73-s + 0.900·79-s + 1/9·81-s + 1.31·83-s + 0.214·87-s − 1.05·89-s + 0.829·93-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.18727452765431, −12.75938288379526, −12.26121338103240, −11.96352594380948, −11.32528537430064, −11.03885995904386, −10.54387092736583, −10.10409939073637, −9.510452979725200, −9.255786997775095, −8.557234789814350, −8.126726980300214, −7.648317856342429, −6.892403570778836, −6.808023212803947, −6.231324631373467, −5.441025976220646, −5.246157610048642, −4.783310618823056, −4.039817017600116, −3.566704199609673, −3.103545947992951, −2.220071622612966, −1.755995409096347, −1.185579361403265, 0, 0,
1.185579361403265, 1.755995409096347, 2.220071622612966, 3.103545947992951, 3.566704199609673, 4.039817017600116, 4.783310618823056, 5.246157610048642, 5.441025976220646, 6.231324631373467, 6.808023212803947, 6.892403570778836, 7.648317856342429, 8.126726980300214, 8.557234789814350, 9.255786997775095, 9.510452979725200, 10.10409939073637, 10.54387092736583, 11.03885995904386, 11.32528537430064, 11.96352594380948, 12.26121338103240, 12.75938288379526, 13.18727452765431
