| L(s) = 1 | + 8.37·3-s + 5·5-s − 9.01·7-s + 43.1·9-s + 3.71·11-s − 40.5·13-s + 41.8·15-s − 91.1·17-s − 77.1·19-s − 75.4·21-s − 23·23-s + 25·25-s + 135.·27-s + 79.6·29-s − 65.4·31-s + 31.1·33-s − 45.0·35-s − 244.·37-s − 339.·39-s − 19.5·41-s − 270.·43-s + 215.·45-s − 183.·47-s − 261.·49-s − 763.·51-s + 283.·53-s + 18.5·55-s + ⋯ |
| L(s) = 1 | + 1.61·3-s + 0.447·5-s − 0.486·7-s + 1.59·9-s + 0.101·11-s − 0.865·13-s + 0.720·15-s − 1.30·17-s − 0.931·19-s − 0.784·21-s − 0.208·23-s + 0.200·25-s + 0.964·27-s + 0.509·29-s − 0.379·31-s + 0.164·33-s − 0.217·35-s − 1.08·37-s − 1.39·39-s − 0.0744·41-s − 0.959·43-s + 0.714·45-s − 0.568·47-s − 0.763·49-s − 2.09·51-s + 0.733·53-s + 0.0455·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 8.37T + 27T^{2} \) |
| 7 | \( 1 + 9.01T + 343T^{2} \) |
| 11 | \( 1 - 3.71T + 1.33e3T^{2} \) |
| 13 | \( 1 + 40.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 91.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 77.1T + 6.85e3T^{2} \) |
| 29 | \( 1 - 79.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 65.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 244.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 19.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 270.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 183.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 283.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 483.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 484.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 44.6T + 3.00e5T^{2} \) |
| 71 | \( 1 - 89.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 346.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 367.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 621.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 652.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 445.T + 9.12e5T^{2} \) |
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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
−8.679891436544081708672621492804, −7.87481390204618791224516272821, −6.95047162009760896198228324633, −6.40109872214970317566198010057, −5.04740546915777077539888666343, −4.16858040114519148983581439578, −3.25880527676246956719751426476, −2.40569322898019646883791774007, −1.77902717131135283391786111924, 0,
1.77902717131135283391786111924, 2.40569322898019646883791774007, 3.25880527676246956719751426476, 4.16858040114519148983581439578, 5.04740546915777077539888666343, 6.40109872214970317566198010057, 6.95047162009760896198228324633, 7.87481390204618791224516272821, 8.679891436544081708672621492804
