| L(s) = 1 | + 2.45·3-s − 5-s − 3.17·7-s + 3.01·9-s − 1.84·11-s + 0.347·13-s − 2.45·15-s + 0.562·17-s − 3.75·19-s − 7.80·21-s + 0.184·23-s + 25-s + 0.0454·27-s − 1.03·29-s + 7.93·31-s − 4.53·33-s + 3.17·35-s + 1.09·37-s + 0.852·39-s + 3.61·41-s + 2.30·43-s − 3.01·45-s + 9.03·47-s + 3.10·49-s + 1.37·51-s + 7.80·53-s + 1.84·55-s + ⋯ |
| L(s) = 1 | + 1.41·3-s − 0.447·5-s − 1.20·7-s + 1.00·9-s − 0.557·11-s + 0.0963·13-s − 0.633·15-s + 0.136·17-s − 0.861·19-s − 1.70·21-s + 0.0384·23-s + 0.200·25-s + 0.00875·27-s − 0.192·29-s + 1.42·31-s − 0.789·33-s + 0.537·35-s + 0.179·37-s + 0.136·39-s + 0.564·41-s + 0.351·43-s − 0.449·45-s + 1.31·47-s + 0.444·49-s + 0.193·51-s + 1.07·53-s + 0.249·55-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
| \(L(1)\) |
\(\approx\) |
\(2.317562292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.317562292\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.45T + 3T^{2} \) |
| 7 | \( 1 + 3.17T + 7T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 13 | \( 1 - 0.347T + 13T^{2} \) |
| 17 | \( 1 - 0.562T + 17T^{2} \) |
| 19 | \( 1 + 3.75T + 19T^{2} \) |
| 23 | \( 1 - 0.184T + 23T^{2} \) |
| 29 | \( 1 + 1.03T + 29T^{2} \) |
| 31 | \( 1 - 7.93T + 31T^{2} \) |
| 37 | \( 1 - 1.09T + 37T^{2} \) |
| 41 | \( 1 - 3.61T + 41T^{2} \) |
| 43 | \( 1 - 2.30T + 43T^{2} \) |
| 47 | \( 1 - 9.03T + 47T^{2} \) |
| 53 | \( 1 - 7.80T + 53T^{2} \) |
| 59 | \( 1 - 5.62T + 59T^{2} \) |
| 61 | \( 1 - 4.83T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 5.05T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 7.27T + 83T^{2} \) |
| 89 | \( 1 + 6.70T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.194665125273730678557283251930, −7.51773551119463973700072768907, −6.79326978442666547199605647219, −6.11889243586164564538174972237, −5.12171646344169075501550651717, −4.01286455340346929877053257863, −3.65842588688367576807951382575, −2.68738829956235205085699672323, −2.32642708113093490504242419636, −0.71610130604449663879847470545,
0.71610130604449663879847470545, 2.32642708113093490504242419636, 2.68738829956235205085699672323, 3.65842588688367576807951382575, 4.01286455340346929877053257863, 5.12171646344169075501550651717, 6.11889243586164564538174972237, 6.79326978442666547199605647219, 7.51773551119463973700072768907, 8.194665125273730678557283251930
