| L(s) = 1 | − 3-s − 2·5-s + 2·7-s − 2·11-s − 7·13-s + 2·15-s + 7·17-s − 6·19-s − 2·21-s − 6·23-s − 7·25-s + 27-s + 29-s − 8·31-s + 2·33-s − 4·35-s − 37-s + 7·39-s − 9·41-s + 6·43-s − 12·47-s + 7·49-s − 7·51-s − 18·53-s + 4·55-s + 6·57-s − 61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.755·7-s − 0.603·11-s − 1.94·13-s + 0.516·15-s + 1.69·17-s − 1.37·19-s − 0.436·21-s − 1.25·23-s − 7/5·25-s + 0.192·27-s + 0.185·29-s − 1.43·31-s + 0.348·33-s − 0.676·35-s − 0.164·37-s + 1.12·39-s − 1.40·41-s + 0.914·43-s − 1.75·47-s + 49-s − 0.980·51-s − 2.47·53-s + 0.539·55-s + 0.794·57-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4492840451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4492840451\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−11.01044362213570543690291797508, −10.39515469773571004074971069278, −10.05239662777259773029015226080, −9.454975354211523756158371760337, −9.414526695253194556839134044414, −8.239053898488920516257274819715, −8.164015062087250201984887037200, −7.65248658266539946976963886744, −7.65234863779502058564052898334, −6.81622352294188804776562072394, −6.37006303992622219680130455723, −5.66166275018837085095380914921, −5.37500875643196997306855864928, −4.70533302244121051840256072489, −4.57622199650985897267776444885, −3.55530915438266595153440864421, −3.43618643805088702417335734839, −2.09145560051354394347550776466, −2.01575817689546221630561996035, −0.36325966543963177653873130146,
0.36325966543963177653873130146, 2.01575817689546221630561996035, 2.09145560051354394347550776466, 3.43618643805088702417335734839, 3.55530915438266595153440864421, 4.57622199650985897267776444885, 4.70533302244121051840256072489, 5.37500875643196997306855864928, 5.66166275018837085095380914921, 6.37006303992622219680130455723, 6.81622352294188804776562072394, 7.65234863779502058564052898334, 7.65248658266539946976963886744, 8.164015062087250201984887037200, 8.239053898488920516257274819715, 9.414526695253194556839134044414, 9.454975354211523756158371760337, 10.05239662777259773029015226080, 10.39515469773571004074971069278, 11.01044362213570543690291797508
