| L(s) = 1 | − 3-s + 5-s + 7-s − 3·11-s + 8·13-s − 15-s + 4·19-s − 21-s − 8·23-s + 5·25-s + 27-s − 6·29-s + 5·31-s + 3·33-s + 35-s − 8·37-s − 8·39-s + 16·41-s + 12·43-s − 10·47-s − 6·49-s − 9·53-s − 3·55-s − 4·57-s + 5·59-s + 10·61-s + 8·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 0.904·11-s + 2.21·13-s − 0.258·15-s + 0.917·19-s − 0.218·21-s − 1.66·23-s + 25-s + 0.192·27-s − 1.11·29-s + 0.898·31-s + 0.522·33-s + 0.169·35-s − 1.31·37-s − 1.28·39-s + 2.49·41-s + 1.82·43-s − 1.45·47-s − 6/7·49-s − 1.23·53-s − 0.404·55-s − 0.529·57-s + 0.650·59-s + 1.28·61-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.173888086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.173888086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T - 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.83942181260212742152509582866, −12.77700785914237666091944282905, −11.99571640242203150594276245619, −11.41031791247393288423130568905, −10.90354637271078615565653728121, −10.89161745293403720966204279901, −10.08768790591763689599556331058, −9.590736515814706101220057799992, −9.000423049924756827261221646976, −8.362824526134932600180029505650, −7.942831106571960374212948670689, −7.42004411163894806630952604738, −6.38018132140269793665373088371, −6.19750591618151433494307400364, −5.51090442417589854718756050162, −5.06532804769675224381579077430, −4.10350619475580917350151538029, −3.46525377428093816798725622350, −2.39487519051647458122973577984, −1.23009030174167561306806809656,
1.23009030174167561306806809656, 2.39487519051647458122973577984, 3.46525377428093816798725622350, 4.10350619475580917350151538029, 5.06532804769675224381579077430, 5.51090442417589854718756050162, 6.19750591618151433494307400364, 6.38018132140269793665373088371, 7.42004411163894806630952604738, 7.942831106571960374212948670689, 8.362824526134932600180029505650, 9.000423049924756827261221646976, 9.590736515814706101220057799992, 10.08768790591763689599556331058, 10.89161745293403720966204279901, 10.90354637271078615565653728121, 11.41031791247393288423130568905, 11.99571640242203150594276245619, 12.77700785914237666091944282905, 12.83942181260212742152509582866
