L(s) = 1 | + 0.936·5-s + 7-s + 4.97·11-s + 1.24·13-s + 5.22·17-s + 5.18·19-s − 2.00·23-s − 4.12·25-s + 6.87·29-s − 5.73·31-s + 0.936·35-s + 9.73·37-s − 11.4·41-s + 9.60·43-s − 1.96·47-s + 49-s − 7.63·53-s + 4.65·55-s + 4.87·59-s − 3.04·61-s + 1.16·65-s − 1.14·67-s − 8.83·71-s + 6.10·73-s + 4.97·77-s − 12.1·79-s − 0.862·83-s + ⋯ |
L(s) = 1 | + 0.418·5-s + 0.377·7-s + 1.49·11-s + 0.345·13-s + 1.26·17-s + 1.18·19-s − 0.418·23-s − 0.824·25-s + 1.27·29-s − 1.02·31-s + 0.158·35-s + 1.59·37-s − 1.79·41-s + 1.46·43-s − 0.287·47-s + 0.142·49-s − 1.04·53-s + 0.628·55-s + 0.635·59-s − 0.389·61-s + 0.144·65-s − 0.140·67-s − 1.04·71-s + 0.714·73-s + 0.566·77-s − 1.36·79-s − 0.0946·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.777035708\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.777035708\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 0.936T + 5T^{2} \) |
| 11 | \( 1 - 4.97T + 11T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 - 5.18T + 19T^{2} \) |
| 23 | \( 1 + 2.00T + 23T^{2} \) |
| 29 | \( 1 - 6.87T + 29T^{2} \) |
| 31 | \( 1 + 5.73T + 31T^{2} \) |
| 37 | \( 1 - 9.73T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 9.60T + 43T^{2} \) |
| 47 | \( 1 + 1.96T + 47T^{2} \) |
| 53 | \( 1 + 7.63T + 53T^{2} \) |
| 59 | \( 1 - 4.87T + 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 + 1.14T + 67T^{2} \) |
| 71 | \( 1 + 8.83T + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 0.862T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 7.57T + 97T^{2} \) |
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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
−8.284382725457915997708510646709, −7.64965516884896199423671676472, −6.87317149838077846177732010880, −6.03378200229950408294695408283, −5.56572494276120712274992818875, −4.55969364648986808497146346440, −3.76815763153321099737253687963, −2.99497208768657424728308767656, −1.71257622018220300533746977130, −1.04203295982649732497629557215,
1.04203295982649732497629557215, 1.71257622018220300533746977130, 2.99497208768657424728308767656, 3.76815763153321099737253687963, 4.55969364648986808497146346440, 5.56572494276120712274992818875, 6.03378200229950408294695408283, 6.87317149838077846177732010880, 7.64965516884896199423671676472, 8.284382725457915997708510646709