| L(s) = 1 | − 0.761·3-s + 15.6·5-s + 7·7-s − 26.4·9-s − 11·11-s − 0.574·13-s − 11.9·15-s − 129.·17-s − 35.3·19-s − 5.33·21-s − 182.·23-s + 119.·25-s + 40.6·27-s − 41.5·29-s + 96.4·31-s + 8.37·33-s + 109.·35-s − 172.·37-s + 0.437·39-s − 385.·41-s + 360.·43-s − 413.·45-s + 72.5·47-s + 49·49-s + 98.8·51-s + 153.·53-s − 172.·55-s + ⋯ |
| L(s) = 1 | − 0.146·3-s + 1.39·5-s + 0.377·7-s − 0.978·9-s − 0.301·11-s − 0.0122·13-s − 0.205·15-s − 1.85·17-s − 0.427·19-s − 0.0554·21-s − 1.65·23-s + 0.957·25-s + 0.290·27-s − 0.266·29-s + 0.558·31-s + 0.0441·33-s + 0.528·35-s − 0.766·37-s + 0.00179·39-s − 1.46·41-s + 1.28·43-s − 1.36·45-s + 0.225·47-s + 0.142·49-s + 0.271·51-s + 0.397·53-s − 0.421·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 0.761T + 27T^{2} \) |
| 5 | \( 1 - 15.6T + 125T^{2} \) |
| 13 | \( 1 + 0.574T + 2.19e3T^{2} \) |
| 17 | \( 1 + 129.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 35.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 182.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 41.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 96.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 172.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 385.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 72.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 153.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 327.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 42.4T + 2.26e5T^{2} \) |
| 67 | \( 1 - 572.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 784.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 452.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 557.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 160.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 395.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.09e3T + 9.12e5T^{2} \) |
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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
−9.858459557290438366342213137677, −8.866051262012561106834287146330, −8.294968706009296684933199447217, −6.87420981841409295357834622395, −6.04046047924669748548694766044, −5.37925943780929913833081469135, −4.24473359222431907117372524547, −2.58561068702553925785434231230, −1.86142104909081744572740522094, 0,
1.86142104909081744572740522094, 2.58561068702553925785434231230, 4.24473359222431907117372524547, 5.37925943780929913833081469135, 6.04046047924669748548694766044, 6.87420981841409295357834622395, 8.294968706009296684933199447217, 8.866051262012561106834287146330, 9.858459557290438366342213137677
