| L(s) = 1 | + 2.40·2-s + 3.79·4-s − 3.50·5-s + 5.09·7-s + 4.32·8-s − 8.42·10-s + 5.50·11-s + 3.91·13-s + 12.2·14-s + 2.81·16-s − 1.79·17-s + 3.82·19-s − 13.2·20-s + 13.2·22-s − 23-s + 7.25·25-s + 9.42·26-s + 19.3·28-s + 29-s − 6.41·31-s − 1.86·32-s − 4.31·34-s − 17.8·35-s − 6.13·37-s + 9.20·38-s − 15.1·40-s − 5.93·41-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.89·4-s − 1.56·5-s + 1.92·7-s + 1.52·8-s − 2.66·10-s + 1.65·11-s + 1.08·13-s + 3.28·14-s + 0.703·16-s − 0.434·17-s + 0.877·19-s − 2.97·20-s + 2.82·22-s − 0.208·23-s + 1.45·25-s + 1.84·26-s + 3.65·28-s + 0.185·29-s − 1.15·31-s − 0.330·32-s − 0.739·34-s − 3.01·35-s − 1.00·37-s + 1.49·38-s − 2.39·40-s − 0.927·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.429056027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.429056027\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 5 | \( 1 + 3.50T + 5T^{2} \) |
| 7 | \( 1 - 5.09T + 7T^{2} \) |
| 11 | \( 1 - 5.50T + 11T^{2} \) |
| 13 | \( 1 - 3.91T + 13T^{2} \) |
| 17 | \( 1 + 1.79T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 31 | \( 1 + 6.41T + 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 + 5.93T + 41T^{2} \) |
| 43 | \( 1 + 3.52T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 1.81T + 53T^{2} \) |
| 59 | \( 1 - 5.37T + 59T^{2} \) |
| 61 | \( 1 - 5.57T + 61T^{2} \) |
| 67 | \( 1 + 7.94T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 6.15T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 2.86T + 89T^{2} \) |
| 97 | \( 1 + 6.80T + 97T^{2} \) |
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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
−7.918618482142261364172859789294, −7.15181436847973962363227607738, −6.69213687816580756734828265884, −5.62691322879446031666970489951, −5.04339535890255672676629331538, −4.23078427971779061594217007922, −3.92606588572523996385693387590, −3.34556264515757436016697398613, −1.96480727281567581239414452935, −1.15809947626540776403194551130,
1.15809947626540776403194551130, 1.96480727281567581239414452935, 3.34556264515757436016697398613, 3.92606588572523996385693387590, 4.23078427971779061594217007922, 5.04339535890255672676629331538, 5.62691322879446031666970489951, 6.69213687816580756734828265884, 7.15181436847973962363227607738, 7.918618482142261364172859789294
