L(s) = 1 | − 2-s − 2.42·3-s + 4-s − 4.26·5-s + 2.42·6-s + 3.45·7-s − 8-s + 2.85·9-s + 4.26·10-s + 1.33·11-s − 2.42·12-s − 3.45·14-s + 10.3·15-s + 16-s − 7.65·17-s − 2.85·18-s − 19-s − 4.26·20-s − 8.35·21-s − 1.33·22-s − 5.48·23-s + 2.42·24-s + 13.1·25-s + 0.342·27-s + 3.45·28-s + 10.4·29-s − 10.3·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.39·3-s + 0.5·4-s − 1.90·5-s + 0.988·6-s + 1.30·7-s − 0.353·8-s + 0.952·9-s + 1.34·10-s + 0.403·11-s − 0.698·12-s − 0.922·14-s + 2.66·15-s + 0.250·16-s − 1.85·17-s − 0.673·18-s − 0.229·19-s − 0.953·20-s − 1.82·21-s − 0.285·22-s − 1.14·23-s + 0.494·24-s + 2.63·25-s + 0.0659·27-s + 0.652·28-s + 1.94·29-s − 1.88·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3334950027\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3334950027\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 2.42T + 3T^{2} \) |
| 5 | \( 1 + 4.26T + 5T^{2} \) |
| 7 | \( 1 - 3.45T + 7T^{2} \) |
| 11 | \( 1 - 1.33T + 11T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 23 | \( 1 + 5.48T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 3.81T + 31T^{2} \) |
| 37 | \( 1 - 0.372T + 37T^{2} \) |
| 41 | \( 1 + 1.97T + 41T^{2} \) |
| 43 | \( 1 - 7.12T + 43T^{2} \) |
| 47 | \( 1 - 9.88T + 47T^{2} \) |
| 53 | \( 1 + 6.96T + 53T^{2} \) |
| 59 | \( 1 + 4.79T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 3.75T + 71T^{2} \) |
| 73 | \( 1 - 0.285T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 - 5.62T + 83T^{2} \) |
| 89 | \( 1 + 3.60T + 89T^{2} \) |
| 97 | \( 1 + 1.75T + 97T^{2} \) |
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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
−8.046369555617982051983487342726, −7.36036239643346744659323707719, −6.71933116515497679712342724404, −6.11219803560948966655557659316, −4.98761685889050141823517954990, −4.46133543034015171955475566517, −3.98515031609390467049771241444, −2.59738165829296035501766939330, −1.36616255891480066072784154385, −0.39000774034106127285054252488,
0.39000774034106127285054252488, 1.36616255891480066072784154385, 2.59738165829296035501766939330, 3.98515031609390467049771241444, 4.46133543034015171955475566517, 4.98761685889050141823517954990, 6.11219803560948966655557659316, 6.71933116515497679712342724404, 7.36036239643346744659323707719, 8.046369555617982051983487342726