| L(s) = 1 | − 1.57·3-s + 5-s − 0.208·7-s − 0.513·9-s − 0.616·11-s − 3.24·13-s − 1.57·15-s + 1.94·17-s + 1.81·19-s + 0.329·21-s − 1.42·23-s + 25-s + 5.54·27-s − 4.26·29-s + 6.21·31-s + 0.971·33-s − 0.208·35-s + 4.31·37-s + 5.11·39-s + 1.53·41-s − 1.84·43-s − 0.513·45-s − 4.09·47-s − 6.95·49-s − 3.07·51-s + 10.8·53-s − 0.616·55-s + ⋯ |
| L(s) = 1 | − 0.910·3-s + 0.447·5-s − 0.0789·7-s − 0.171·9-s − 0.185·11-s − 0.899·13-s − 0.407·15-s + 0.472·17-s + 0.416·19-s + 0.0718·21-s − 0.297·23-s + 0.200·25-s + 1.06·27-s − 0.792·29-s + 1.11·31-s + 0.169·33-s − 0.0352·35-s + 0.709·37-s + 0.818·39-s + 0.239·41-s − 0.281·43-s − 0.0766·45-s − 0.597·47-s − 0.993·49-s − 0.430·51-s + 1.48·53-s − 0.0831·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 + 1.57T + 3T^{2} \) |
| 7 | \( 1 + 0.208T + 7T^{2} \) |
| 11 | \( 1 + 0.616T + 11T^{2} \) |
| 13 | \( 1 + 3.24T + 13T^{2} \) |
| 17 | \( 1 - 1.94T + 17T^{2} \) |
| 19 | \( 1 - 1.81T + 19T^{2} \) |
| 23 | \( 1 + 1.42T + 23T^{2} \) |
| 29 | \( 1 + 4.26T + 29T^{2} \) |
| 31 | \( 1 - 6.21T + 31T^{2} \) |
| 37 | \( 1 - 4.31T + 37T^{2} \) |
| 41 | \( 1 - 1.53T + 41T^{2} \) |
| 43 | \( 1 + 1.84T + 43T^{2} \) |
| 47 | \( 1 + 4.09T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 8.06T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 5.24T + 67T^{2} \) |
| 71 | \( 1 - 8.10T + 71T^{2} \) |
| 73 | \( 1 + 1.11T + 73T^{2} \) |
| 79 | \( 1 + 7.47T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 9.40T + 97T^{2} \) |
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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
−7.65869279783766180312126302724, −6.88552778896417701341207069578, −6.21180055267583860307963938683, −5.53669158793204083770866471388, −5.06834095802311079871838034949, −4.23101785026797301155178797918, −3.10168230642286775285003098760, −2.36037426181602344169641644563, −1.15099172947200930871463421380, 0,
1.15099172947200930871463421380, 2.36037426181602344169641644563, 3.10168230642286775285003098760, 4.23101785026797301155178797918, 5.06834095802311079871838034949, 5.53669158793204083770866471388, 6.21180055267583860307963938683, 6.88552778896417701341207069578, 7.65869279783766180312126302724
