| L(s) = 1 | − 2.56·3-s + 3.59·9-s − 1.56·11-s + 5.56·13-s + 7.16·17-s + 3.16·19-s + 5.73·23-s − 1.53·27-s + 1.96·29-s + 0.969·31-s + 4.03·33-s − 6.70·37-s − 14.3·39-s − 8.87·41-s − 4.59·43-s + 0.401·47-s − 18.4·51-s + 9.53·53-s − 8.13·57-s − 4.56·59-s + 15.3·61-s + 0.862·67-s − 14.7·69-s − 12.1·71-s + 4·73-s − 12.8·79-s − 6.84·81-s + ⋯ |
| L(s) = 1 | − 1.48·3-s + 1.19·9-s − 0.473·11-s + 1.54·13-s + 1.73·17-s + 0.726·19-s + 1.19·23-s − 0.296·27-s + 0.365·29-s + 0.174·31-s + 0.701·33-s − 1.10·37-s − 2.29·39-s − 1.38·41-s − 0.701·43-s + 0.0584·47-s − 2.57·51-s + 1.31·53-s − 1.07·57-s − 0.594·59-s + 1.95·61-s + 0.105·67-s − 1.77·69-s − 1.44·71-s + 0.468·73-s − 1.44·79-s − 0.760·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.304168893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.304168893\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 5.56T + 13T^{2} \) |
| 17 | \( 1 - 7.16T + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 - 1.96T + 29T^{2} \) |
| 31 | \( 1 - 0.969T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 + 8.87T + 41T^{2} \) |
| 43 | \( 1 + 4.59T + 43T^{2} \) |
| 47 | \( 1 - 0.401T + 47T^{2} \) |
| 53 | \( 1 - 9.53T + 53T^{2} \) |
| 59 | \( 1 + 4.56T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 - 0.862T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 5.59T + 89T^{2} \) |
| 97 | \( 1 - 0.233T + 97T^{2} \) |
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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
−8.272895037808005411095673066868, −7.32978423627467424013688552157, −6.72963229750414867187342172892, −5.94401848486141309948851984287, −5.37219531680335811788711992720, −4.94311231927924970412377331296, −3.72925792048554045041174889215, −3.08012350977213722019998616421, −1.45735418037543622038551920924, −0.75680612626105744959477088505,
0.75680612626105744959477088505, 1.45735418037543622038551920924, 3.08012350977213722019998616421, 3.72925792048554045041174889215, 4.94311231927924970412377331296, 5.37219531680335811788711992720, 5.94401848486141309948851984287, 6.72963229750414867187342172892, 7.32978423627467424013688552157, 8.272895037808005411095673066868
