| L(s) = 1 | + 3-s − 5-s + 4.20·7-s + 9-s + 2.75·11-s − 0.753·13-s − 15-s − 4.96·17-s + 4.75·19-s + 4.20·21-s + 23-s + 25-s + 27-s + 4.96·29-s − 0.209·31-s + 2.75·33-s − 4.20·35-s − 5.71·37-s − 0.753·39-s − 9.38·41-s + 12.4·43-s − 45-s + 7.17·47-s + 10.7·49-s − 4.96·51-s − 9.38·53-s − 2.75·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.59·7-s + 0.333·9-s + 0.830·11-s − 0.208·13-s − 0.258·15-s − 1.20·17-s + 1.09·19-s + 0.918·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.921·29-s − 0.0375·31-s + 0.479·33-s − 0.711·35-s − 0.939·37-s − 0.120·39-s − 1.46·41-s + 1.89·43-s − 0.149·45-s + 1.04·47-s + 1.53·49-s − 0.694·51-s − 1.28·53-s − 0.371·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.393767967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.393767967\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4.20T + 7T^{2} \) |
| 11 | \( 1 - 2.75T + 11T^{2} \) |
| 13 | \( 1 + 0.753T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 19 | \( 1 - 4.75T + 19T^{2} \) |
| 29 | \( 1 - 4.96T + 29T^{2} \) |
| 31 | \( 1 + 0.209T + 31T^{2} \) |
| 37 | \( 1 + 5.71T + 37T^{2} \) |
| 41 | \( 1 + 9.38T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 - 7.17T + 47T^{2} \) |
| 53 | \( 1 + 9.38T + 53T^{2} \) |
| 59 | \( 1 + 4.96T + 59T^{2} \) |
| 61 | \( 1 + 5.17T + 61T^{2} \) |
| 67 | \( 1 + 0.209T + 67T^{2} \) |
| 71 | \( 1 - 9.38T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 2.41T + 79T^{2} \) |
| 83 | \( 1 - 5.45T + 83T^{2} \) |
| 89 | \( 1 - 4.91T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−9.283847332152058150982953246763, −8.784524163453622519273792604468, −7.973207165389061169146942735373, −7.36106031408583522974286330676, −6.46511903164147711739279338069, −5.10597839717954388070057587370, −4.51317189637090511227030872574, −3.56670292035907263607382784903, −2.29904300338527129077219014903, −1.22767801999255964670496877028,
1.22767801999255964670496877028, 2.29904300338527129077219014903, 3.56670292035907263607382784903, 4.51317189637090511227030872574, 5.10597839717954388070057587370, 6.46511903164147711739279338069, 7.36106031408583522974286330676, 7.973207165389061169146942735373, 8.784524163453622519273792604468, 9.283847332152058150982953246763
