| L(s) = 1 | − 3-s − 5-s + 3.73·7-s + 9-s − 4.84·11-s − 2.84·13-s + 15-s + 0.890·17-s − 6.84·19-s − 3.73·21-s − 23-s + 25-s − 27-s − 0.890·29-s − 7.73·31-s + 4.84·33-s − 3.73·35-s − 1.95·37-s + 2.84·39-s + 12.3·41-s + 3.47·43-s − 45-s + 6.62·47-s + 6.95·49-s − 0.890·51-s + 12.3·53-s + 4.84·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.41·7-s + 0.333·9-s − 1.46·11-s − 0.788·13-s + 0.258·15-s + 0.216·17-s − 1.57·19-s − 0.815·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s − 0.165·29-s − 1.38·31-s + 0.843·33-s − 0.631·35-s − 0.321·37-s + 0.455·39-s + 1.93·41-s + 0.529·43-s − 0.149·45-s + 0.966·47-s + 0.993·49-s − 0.124·51-s + 1.69·53-s + 0.653·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.115420533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.115420533\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 3.73T + 7T^{2} \) |
| 11 | \( 1 + 4.84T + 11T^{2} \) |
| 13 | \( 1 + 2.84T + 13T^{2} \) |
| 17 | \( 1 - 0.890T + 17T^{2} \) |
| 19 | \( 1 + 6.84T + 19T^{2} \) |
| 29 | \( 1 + 0.890T + 29T^{2} \) |
| 31 | \( 1 + 7.73T + 31T^{2} \) |
| 37 | \( 1 + 1.95T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 - 3.47T + 43T^{2} \) |
| 47 | \( 1 - 6.62T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 0.890T + 59T^{2} \) |
| 61 | \( 1 - 8.62T + 61T^{2} \) |
| 67 | \( 1 + 7.73T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 4.58T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
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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
−8.057962229421811608305950691485, −7.49545686675649568097929187760, −6.93146970404755496431654384597, −5.64636316829900980952359678357, −5.39615097898362078985309661355, −4.49411956088056634368087831401, −4.02008612687333361725807941306, −2.56927071739488823782904089074, −1.96437124743490682258231862530, −0.56402317618019820284477534291,
0.56402317618019820284477534291, 1.96437124743490682258231862530, 2.56927071739488823782904089074, 4.02008612687333361725807941306, 4.49411956088056634368087831401, 5.39615097898362078985309661355, 5.64636316829900980952359678357, 6.93146970404755496431654384597, 7.49545686675649568097929187760, 8.057962229421811608305950691485
