| L(s) = 1 | − 2-s − 3.16·3-s + 4-s + 3.16·6-s − 7-s − 8-s + 7.00·9-s − 11-s − 3.16·12-s + 0.162·13-s + 14-s + 16-s − 0.837·17-s − 7.00·18-s − 19-s + 3.16·21-s + 22-s + 2.16·23-s + 3.16·24-s − 0.162·26-s − 12.6·27-s − 28-s + 3·29-s − 7·31-s − 32-s + 3.16·33-s + 0.837·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.82·3-s + 0.5·4-s + 1.29·6-s − 0.377·7-s − 0.353·8-s + 2.33·9-s − 0.301·11-s − 0.912·12-s + 0.0450·13-s + 0.267·14-s + 0.250·16-s − 0.203·17-s − 1.64·18-s − 0.229·19-s + 0.690·21-s + 0.213·22-s + 0.450·23-s + 0.645·24-s − 0.0318·26-s − 2.43·27-s − 0.188·28-s + 0.557·29-s − 1.25·31-s − 0.176·32-s + 0.550·33-s + 0.143·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 3.16T + 3T^{2} \) |
| 13 | \( 1 - 0.162T + 13T^{2} \) |
| 17 | \( 1 + 0.837T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 2.16T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 6.32T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 0.162T + 43T^{2} \) |
| 47 | \( 1 + 4.32T + 47T^{2} \) |
| 53 | \( 1 + 0.837T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 7.48T + 67T^{2} \) |
| 71 | \( 1 - 3.83T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 3.16T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 2.16T + 89T^{2} \) |
| 97 | \( 1 + 5.83T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.998958537171458278270032504167, −7.16417176533056613786548387251, −6.67360456373116796699651099417, −5.95840509297329145588445166871, −5.32097263088965162929537013877, −4.54256071403937230807766755617, −3.49786090844016280905289570181, −2.12901011596195684042037586484, −0.984562994382800583068742871958, 0,
0.984562994382800583068742871958, 2.12901011596195684042037586484, 3.49786090844016280905289570181, 4.54256071403937230807766755617, 5.32097263088965162929537013877, 5.95840509297329145588445166871, 6.67360456373116796699651099417, 7.16417176533056613786548387251, 7.998958537171458278270032504167
