| L(s) = 1 | − 1.36·3-s + 2.36·5-s + 1.37·7-s − 1.13·9-s + 11-s + 2.35·13-s − 3.22·15-s + 7.93·17-s + 1.27·19-s − 1.88·21-s + 2.63·23-s + 0.591·25-s + 5.64·27-s + 7.48·29-s + 3.45·31-s − 1.36·33-s + 3.26·35-s − 6.90·37-s − 3.20·39-s − 9.01·41-s + 11.3·43-s − 2.69·45-s − 3.33·47-s − 5.09·49-s − 10.8·51-s − 8.40·53-s + 2.36·55-s + ⋯ |
| L(s) = 1 | − 0.787·3-s + 1.05·5-s + 0.521·7-s − 0.379·9-s + 0.301·11-s + 0.652·13-s − 0.833·15-s + 1.92·17-s + 0.291·19-s − 0.410·21-s + 0.549·23-s + 0.118·25-s + 1.08·27-s + 1.38·29-s + 0.619·31-s − 0.237·33-s + 0.551·35-s − 1.13·37-s − 0.513·39-s − 1.40·41-s + 1.72·43-s − 0.401·45-s − 0.487·47-s − 0.728·49-s − 1.51·51-s − 1.15·53-s + 0.318·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.287023673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.287023673\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 + 1.36T + 3T^{2} \) |
| 5 | \( 1 - 2.36T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 13 | \( 1 - 2.35T + 13T^{2} \) |
| 17 | \( 1 - 7.93T + 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 - 2.63T + 23T^{2} \) |
| 29 | \( 1 - 7.48T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 + 6.90T + 37T^{2} \) |
| 41 | \( 1 + 9.01T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 3.33T + 47T^{2} \) |
| 53 | \( 1 + 8.40T + 53T^{2} \) |
| 59 | \( 1 - 7.55T + 59T^{2} \) |
| 61 | \( 1 - 7.73T + 61T^{2} \) |
| 67 | \( 1 + 6.85T + 67T^{2} \) |
| 71 | \( 1 + 1.14T + 71T^{2} \) |
| 73 | \( 1 + 0.987T + 73T^{2} \) |
| 79 | \( 1 - 3.10T + 79T^{2} \) |
| 83 | \( 1 + 0.719T + 83T^{2} \) |
| 89 | \( 1 + 1.42T + 89T^{2} \) |
| 97 | \( 1 + 0.959T + 97T^{2} \) |
| 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
−8.216035091174264863590683047967, −7.25078204849748427712674138787, −6.40814984928735802976147430468, −5.93008846689046040243448135372, −5.28648628247289231011350630715, −4.80408188061921252639487116441, −3.54992346233676357203728707999, −2.80254624408308274705147016820, −1.59115034852898865299055000507, −0.911227785885863249743168882827,
0.911227785885863249743168882827, 1.59115034852898865299055000507, 2.80254624408308274705147016820, 3.54992346233676357203728707999, 4.80408188061921252639487116441, 5.28648628247289231011350630715, 5.93008846689046040243448135372, 6.40814984928735802976147430468, 7.25078204849748427712674138787, 8.216035091174264863590683047967
