L(s) = 1 | − 3-s + 5-s − 4.67·7-s + 9-s − 5.81·11-s − 2.67·13-s − 15-s − 6.48·17-s + 19-s + 4.67·21-s − 8.20·23-s + 25-s − 27-s − 3.81·29-s + 6.20·31-s + 5.81·33-s − 4.67·35-s − 12.0·37-s + 2.67·39-s + 0.183·41-s + 6.96·43-s + 45-s + 4.20·47-s + 14.8·49-s + 6.48·51-s + 10.7·53-s − 5.81·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.76·7-s + 0.333·9-s − 1.75·11-s − 0.741·13-s − 0.258·15-s − 1.57·17-s + 0.229·19-s + 1.01·21-s − 1.71·23-s + 0.200·25-s − 0.192·27-s − 0.708·29-s + 1.11·31-s + 1.01·33-s − 0.789·35-s − 1.97·37-s + 0.427·39-s + 0.0286·41-s + 1.06·43-s + 0.149·45-s + 0.612·47-s + 2.11·49-s + 0.908·51-s + 1.48·53-s − 0.784·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2890044110\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2890044110\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 4.67T + 7T^{2} \) |
| 11 | \( 1 + 5.81T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 + 6.48T + 17T^{2} \) |
| 23 | \( 1 + 8.20T + 23T^{2} \) |
| 29 | \( 1 + 3.81T + 29T^{2} \) |
| 31 | \( 1 - 6.20T + 31T^{2} \) |
| 37 | \( 1 + 12.0T + 37T^{2} \) |
| 41 | \( 1 - 0.183T + 41T^{2} \) |
| 43 | \( 1 - 6.96T + 43T^{2} \) |
| 47 | \( 1 - 4.20T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 7.05T + 59T^{2} \) |
| 61 | \( 1 - 5.14T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 4.28T + 73T^{2} \) |
| 79 | \( 1 - 6.28T + 79T^{2} \) |
| 83 | \( 1 + 1.14T + 83T^{2} \) |
| 89 | \( 1 + 3.81T + 89T^{2} \) |
| 97 | \( 1 + 6.67T + 97T^{2} \) |
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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
−8.379907558902458640987664150460, −7.30925108502432286422944268858, −6.92077826475855362974112409661, −5.98197262982819733894319800287, −5.63434217598914083511142956427, −4.68892666708127554389538099870, −3.80304572312700733500898939915, −2.70167708746831260877864579500, −2.20153256751236094641287620428, −0.28061460026514670374803679791,
0.28061460026514670374803679791, 2.20153256751236094641287620428, 2.70167708746831260877864579500, 3.80304572312700733500898939915, 4.68892666708127554389538099870, 5.63434217598914083511142956427, 5.98197262982819733894319800287, 6.92077826475855362974112409661, 7.30925108502432286422944268858, 8.379907558902458640987664150460