| L(s) = 1 | + 84·3-s − 82·5-s + 456·7-s + 4.86e3·9-s + 2.52e3·11-s − 1.07e4·13-s − 6.88e3·15-s − 1.11e4·17-s − 4.12e3·19-s + 3.83e4·21-s − 8.17e4·23-s − 7.14e4·25-s + 2.25e5·27-s + 9.97e4·29-s + 4.04e4·31-s + 2.12e5·33-s − 3.73e4·35-s − 4.19e5·37-s − 9.05e5·39-s + 1.41e5·41-s + 6.90e5·43-s − 3.99e5·45-s + 6.82e5·47-s − 6.15e5·49-s − 9.36e5·51-s + 1.81e6·53-s − 2.06e5·55-s + ⋯ |
| L(s) = 1 | + 1.79·3-s − 0.293·5-s + 0.502·7-s + 2.22·9-s + 0.571·11-s − 1.36·13-s − 0.526·15-s − 0.550·17-s − 0.137·19-s + 0.902·21-s − 1.40·23-s − 0.913·25-s + 2.20·27-s + 0.759·29-s + 0.244·31-s + 1.02·33-s − 0.147·35-s − 1.36·37-s − 2.44·39-s + 0.320·41-s + 1.32·43-s − 0.653·45-s + 0.958·47-s − 0.747·49-s − 0.988·51-s + 1.67·53-s − 0.167·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.478836145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.478836145\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 28 p T + p^{7} T^{2} \) |
| 5 | \( 1 + 82 T + p^{7} T^{2} \) |
| 7 | \( 1 - 456 T + p^{7} T^{2} \) |
| 11 | \( 1 - 2524 T + p^{7} T^{2} \) |
| 13 | \( 1 + 10778 T + p^{7} T^{2} \) |
| 17 | \( 1 + 11150 T + p^{7} T^{2} \) |
| 19 | \( 1 + 4124 T + p^{7} T^{2} \) |
| 23 | \( 1 + 81704 T + p^{7} T^{2} \) |
| 29 | \( 1 - 99798 T + p^{7} T^{2} \) |
| 31 | \( 1 - 40480 T + p^{7} T^{2} \) |
| 37 | \( 1 + 419442 T + p^{7} T^{2} \) |
| 41 | \( 1 - 141402 T + p^{7} T^{2} \) |
| 43 | \( 1 - 690428 T + p^{7} T^{2} \) |
| 47 | \( 1 - 682032 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1813118 T + p^{7} T^{2} \) |
| 59 | \( 1 - 966028 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1887670 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2965868 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2548232 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1680326 T + p^{7} T^{2} \) |
| 79 | \( 1 + 4038064 T + p^{7} T^{2} \) |
| 83 | \( 1 - 5385764 T + p^{7} T^{2} \) |
| 89 | \( 1 + 6473046 T + p^{7} T^{2} \) |
| 97 | \( 1 + 6065758 T + p^{7} T^{2} \) |
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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
−17.66721966277571562453584863051, −15.73636509840895306549251364077, −14.64093820418034096242442089365, −13.81730268455133328447094630327, −12.16596542212007107458239843125, −9.931569659816554879366713212046, −8.589870178199185874217335919010, −7.38865751319070296706883262229, −4.13265982093190446209974157805, −2.23219012322178757280172492286,
2.23219012322178757280172492286, 4.13265982093190446209974157805, 7.38865751319070296706883262229, 8.589870178199185874217335919010, 9.931569659816554879366713212046, 12.16596542212007107458239843125, 13.81730268455133328447094630327, 14.64093820418034096242442089365, 15.73636509840895306549251364077, 17.66721966277571562453584863051
