L(s) = 1 | + 1.36·3-s − 2.30·5-s − 7-s − 1.12·9-s − 5.06·11-s − 1.83·13-s − 3.14·15-s − 4.12·17-s + 6.81·19-s − 1.36·21-s − 4.65·23-s + 0.298·25-s − 5.64·27-s − 6.94·29-s + 0.486·31-s − 6.93·33-s + 2.30·35-s − 9.19·37-s − 2.51·39-s − 0.108·41-s − 2.67·43-s + 2.59·45-s + 7.72·47-s + 49-s − 5.64·51-s − 7.65·53-s + 11.6·55-s + ⋯ |
L(s) = 1 | + 0.789·3-s − 1.02·5-s − 0.377·7-s − 0.375·9-s − 1.52·11-s − 0.509·13-s − 0.813·15-s − 1.00·17-s + 1.56·19-s − 0.298·21-s − 0.970·23-s + 0.0596·25-s − 1.08·27-s − 1.29·29-s + 0.0873·31-s − 1.20·33-s + 0.389·35-s − 1.51·37-s − 0.402·39-s − 0.0170·41-s − 0.408·43-s + 0.386·45-s + 1.12·47-s + 0.142·49-s − 0.790·51-s − 1.05·53-s + 1.57·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5984840141\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5984840141\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 89 | \( 1 + T \) |
good | 3 | \( 1 - 1.36T + 3T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 11 | \( 1 + 5.06T + 11T^{2} \) |
| 13 | \( 1 + 1.83T + 13T^{2} \) |
| 17 | \( 1 + 4.12T + 17T^{2} \) |
| 19 | \( 1 - 6.81T + 19T^{2} \) |
| 23 | \( 1 + 4.65T + 23T^{2} \) |
| 29 | \( 1 + 6.94T + 29T^{2} \) |
| 31 | \( 1 - 0.486T + 31T^{2} \) |
| 37 | \( 1 + 9.19T + 37T^{2} \) |
| 41 | \( 1 + 0.108T + 41T^{2} \) |
| 43 | \( 1 + 2.67T + 43T^{2} \) |
| 47 | \( 1 - 7.72T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 - 3.17T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 8.76T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 9.37T + 79T^{2} \) |
| 83 | \( 1 - 5.06T + 83T^{2} \) |
| 97 | \( 1 - 6.09T + 97T^{2} \) |
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
−7.70788226132795440499674026507, −7.33835800311649319620083697752, −6.39362939912056778814820141577, −5.38679458391685084825327886188, −5.03384833001976725284109496105, −3.84274711050396883862992759131, −3.49462691133101397089091858034, −2.63327059883335556319016025952, −2.03874917250952126919087671724, −0.32286605047799383637080267784,
0.32286605047799383637080267784, 2.03874917250952126919087671724, 2.63327059883335556319016025952, 3.49462691133101397089091858034, 3.84274711050396883862992759131, 5.03384833001976725284109496105, 5.38679458391685084825327886188, 6.39362939912056778814820141577, 7.33835800311649319620083697752, 7.70788226132795440499674026507