| L(s) = 1 | − 2-s + 4-s − 8-s − 5·9-s − 12·11-s + 10·13-s + 16-s + 6·17-s + 5·18-s + 19-s + 12·22-s − 10·25-s − 10·26-s + 18·29-s − 8·31-s − 32-s − 6·34-s − 5·36-s + 4·37-s − 38-s + 16·43-s − 12·44-s − 13·49-s + 10·50-s + 10·52-s − 6·53-s − 18·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 5/3·9-s − 3.61·11-s + 2.77·13-s + 1/4·16-s + 1.45·17-s + 1.17·18-s + 0.229·19-s + 2.55·22-s − 2·25-s − 1.96·26-s + 3.34·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s − 5/6·36-s + 0.657·37-s − 0.162·38-s + 2.43·43-s − 1.80·44-s − 1.85·49-s + 1.41·50-s + 1.38·52-s − 0.824·53-s − 2.36·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.895349544747219426101327340569, −7.73591843742988285704805588989, −7.55387598440628310474150895416, −6.51678176774053671258970671852, −5.84120552555487673973212425580, −5.81027585652647529822118508915, −5.68309337660531866369187028825, −4.89812282859220086788968346022, −4.28044972533877048207987886223, −3.31668847110395308659409555660, −2.99432139105097413830284496966, −2.79906366636498682136535089092, −1.91153416476236487854080031114, −0.932303105265167456620735982403, 0,
0.932303105265167456620735982403, 1.91153416476236487854080031114, 2.79906366636498682136535089092, 2.99432139105097413830284496966, 3.31668847110395308659409555660, 4.28044972533877048207987886223, 4.89812282859220086788968346022, 5.68309337660531866369187028825, 5.81027585652647529822118508915, 5.84120552555487673973212425580, 6.51678176774053671258970671852, 7.55387598440628310474150895416, 7.73591843742988285704805588989, 7.895349544747219426101327340569
