| L(s) = 1 | + 12·3-s − 54·5-s − 99·9-s + 540·11-s + 418·13-s − 648·15-s − 594·17-s − 836·19-s − 4.10e3·23-s − 209·25-s − 4.10e3·27-s − 594·29-s − 4.25e3·31-s + 6.48e3·33-s − 298·37-s + 5.01e3·39-s − 1.72e4·41-s − 1.21e4·43-s + 5.34e3·45-s + 1.29e3·47-s − 7.12e3·51-s + 1.94e4·53-s − 2.91e4·55-s − 1.00e4·57-s + 7.66e3·59-s + 3.47e4·61-s − 2.25e4·65-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 0.965·5-s − 0.407·9-s + 1.34·11-s + 0.685·13-s − 0.743·15-s − 0.498·17-s − 0.531·19-s − 1.61·23-s − 0.0668·25-s − 1.08·27-s − 0.131·29-s − 0.795·31-s + 1.03·33-s − 0.0357·37-s + 0.528·39-s − 1.60·41-s − 0.997·43-s + 0.393·45-s + 0.0855·47-s − 0.383·51-s + 0.953·53-s − 1.29·55-s − 0.408·57-s + 0.286·59-s + 1.19·61-s − 0.662·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 4 p T + p^{5} T^{2} \) |
| 5 | \( 1 + 54 T + p^{5} T^{2} \) |
| 11 | \( 1 - 540 T + p^{5} T^{2} \) |
| 13 | \( 1 - 418 T + p^{5} T^{2} \) |
| 17 | \( 1 + 594 T + p^{5} T^{2} \) |
| 19 | \( 1 + 44 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 4104 T + p^{5} T^{2} \) |
| 29 | \( 1 + 594 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4256 T + p^{5} T^{2} \) |
| 37 | \( 1 + 298 T + p^{5} T^{2} \) |
| 41 | \( 1 + 17226 T + p^{5} T^{2} \) |
| 43 | \( 1 + 12100 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1296 T + p^{5} T^{2} \) |
| 53 | \( 1 - 19494 T + p^{5} T^{2} \) |
| 59 | \( 1 - 7668 T + p^{5} T^{2} \) |
| 61 | \( 1 - 34738 T + p^{5} T^{2} \) |
| 67 | \( 1 - 21812 T + p^{5} T^{2} \) |
| 71 | \( 1 + 46872 T + p^{5} T^{2} \) |
| 73 | \( 1 + 67562 T + p^{5} T^{2} \) |
| 79 | \( 1 + 76912 T + p^{5} T^{2} \) |
| 83 | \( 1 + 67716 T + p^{5} T^{2} \) |
| 89 | \( 1 + 29754 T + p^{5} T^{2} \) |
| 97 | \( 1 - 122398 T + p^{5} T^{2} \) |
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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
−11.44685186150128413106762335950, −10.03503985661933840888228477404, −8.773279524208492506143451817734, −8.353785535572860198352928879109, −7.12887503936441446215887311078, −5.96123271535316744908512171002, −4.14992916287121959950978852690, −3.49474016109518129157168649396, −1.85204006631698210774758760246, 0,
1.85204006631698210774758760246, 3.49474016109518129157168649396, 4.14992916287121959950978852690, 5.96123271535316744908512171002, 7.12887503936441446215887311078, 8.353785535572860198352928879109, 8.773279524208492506143451817734, 10.03503985661933840888228477404, 11.44685186150128413106762335950
