L(s) = 1 | − 2.20·2-s − 2.83·3-s + 2.85·4-s + 6.25·6-s + 7-s − 1.87·8-s + 5.05·9-s + 4.31·11-s − 8.09·12-s − 1.72·13-s − 2.20·14-s − 1.57·16-s + 2.56·17-s − 11.1·18-s + 2.10·19-s − 2.83·21-s − 9.49·22-s + 23-s + 5.31·24-s + 3.79·26-s − 5.83·27-s + 2.85·28-s + 10.3·29-s + 1.07·31-s + 7.21·32-s − 12.2·33-s − 5.65·34-s + ⋯ |
L(s) = 1 | − 1.55·2-s − 1.63·3-s + 1.42·4-s + 2.55·6-s + 0.377·7-s − 0.662·8-s + 1.68·9-s + 1.29·11-s − 2.33·12-s − 0.478·13-s − 0.588·14-s − 0.393·16-s + 0.623·17-s − 2.62·18-s + 0.483·19-s − 0.619·21-s − 2.02·22-s + 0.208·23-s + 1.08·24-s + 0.744·26-s − 1.12·27-s + 0.538·28-s + 1.91·29-s + 0.192·31-s + 1.27·32-s − 2.12·33-s − 0.970·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6406436727\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6406436727\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 3 | \( 1 + 2.83T + 3T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 13 | \( 1 + 1.72T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 2.10T + 19T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 1.07T + 31T^{2} \) |
| 37 | \( 1 - 8.45T + 37T^{2} \) |
| 41 | \( 1 - 1.65T + 41T^{2} \) |
| 43 | \( 1 - 8.34T + 43T^{2} \) |
| 47 | \( 1 - 4.13T + 47T^{2} \) |
| 53 | \( 1 + 4.22T + 53T^{2} \) |
| 59 | \( 1 - 5.12T + 59T^{2} \) |
| 61 | \( 1 - 5.11T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 9.34T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 1.72T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 5.63T + 89T^{2} \) |
| 97 | \( 1 + 7.16T + 97T^{2} \) |
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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
−8.433937791735963401910625328247, −7.75363439420175678165242163786, −6.91470618961952454047468519047, −6.53209858247631513473904609349, −5.68351427863638873652250485662, −4.82475394169349251840004213012, −4.07327328803306893825168440218, −2.50126757113510547758863290326, −1.16223626168004664591861266815, −0.810390193992491724368693576403,
0.810390193992491724368693576403, 1.16223626168004664591861266815, 2.50126757113510547758863290326, 4.07327328803306893825168440218, 4.82475394169349251840004213012, 5.68351427863638873652250485662, 6.53209858247631513473904609349, 6.91470618961952454047468519047, 7.75363439420175678165242163786, 8.433937791735963401910625328247