| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s + 2·11-s + 13-s − 15-s + 4·17-s + 2·19-s − 21-s + 23-s + 25-s + 5·27-s + 3·29-s + 9·31-s − 2·33-s + 35-s − 2·37-s − 39-s − 5·41-s + 4·43-s − 2·45-s + 47-s + 49-s − 4·51-s − 6·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s − 0.258·15-s + 0.970·17-s + 0.458·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s + 1.61·31-s − 0.348·33-s + 0.169·35-s − 0.328·37-s − 0.160·39-s − 0.780·41-s + 0.609·43-s − 0.298·45-s + 0.145·47-s + 1/7·49-s − 0.560·51-s − 0.824·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−14.78906809211855, −14.09971506946526, −13.78814320818162, −13.41153630937212, −12.44795239325308, −12.08897831324747, −11.79701125642517, −11.17898777296393, −10.57739773088587, −10.27635758430935, −9.489446806303282, −9.128373571227968, −8.396603403903955, −8.027953223790408, −7.340771403803016, −6.530132149424786, −6.296441672815258, −5.590052096332618, −5.170598402769033, −4.565617679349487, −3.852902858632216, −2.986955528082744, −2.659635364119125, −1.406398010978206, −1.159719235758855, 0,
1.159719235758855, 1.406398010978206, 2.659635364119125, 2.986955528082744, 3.852902858632216, 4.565617679349487, 5.170598402769033, 5.590052096332618, 6.296441672815258, 6.530132149424786, 7.340771403803016, 8.027953223790408, 8.396603403903955, 9.128373571227968, 9.489446806303282, 10.27635758430935, 10.57739773088587, 11.17898777296393, 11.79701125642517, 12.08897831324747, 12.44795239325308, 13.41153630937212, 13.78814320818162, 14.09971506946526, 14.78906809211855
