L(s) = 1 | + 6·7-s + 8·13-s − 6·17-s + 12·19-s − 12·23-s + 6·25-s + 4·29-s − 12·31-s + 12·37-s − 8·41-s + 6·47-s + 18·49-s − 24·59-s + 12·67-s + 18·71-s − 18·79-s − 9·81-s + 10·89-s + 48·91-s + 10·97-s + 24·107-s − 8·109-s − 4·113-s − 36·119-s + 127-s + 131-s + 72·133-s + ⋯ |
L(s) = 1 | + 2.26·7-s + 2.21·13-s − 1.45·17-s + 2.75·19-s − 2.50·23-s + 6/5·25-s + 0.742·29-s − 2.15·31-s + 1.97·37-s − 1.24·41-s + 0.875·47-s + 18/7·49-s − 3.12·59-s + 1.46·67-s + 2.13·71-s − 2.02·79-s − 81-s + 1.05·89-s + 5.03·91-s + 1.01·97-s + 2.32·107-s − 0.766·109-s − 0.376·113-s − 3.30·119-s + 0.0887·127-s + 0.0873·131-s + 6.24·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1721344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1721344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.702838147\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.702838147\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 41 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.615517958435247271688481508354, −9.501418085284460779059325874929, −8.937800276168340691835853866026, −8.388318984386349664741614546375, −8.379895306690642539680548967067, −7.927858138553774775359398639272, −7.39857562650960081117665772677, −7.23575387616021305799087228683, −6.46383618078303211486933142084, −5.93101537184387399017731578862, −5.80039929502246160298895306554, −5.16514684409127151400237530347, −4.67745359171453829973301412698, −4.45618174639019371561735875075, −3.67525488885812910803106833635, −3.50945493428374834511232491968, −2.59939262525778985380027188086, −1.85933774204870186897596012079, −1.51171402093699181445155323166, −0.895367555727343382170894714370,
0.895367555727343382170894714370, 1.51171402093699181445155323166, 1.85933774204870186897596012079, 2.59939262525778985380027188086, 3.50945493428374834511232491968, 3.67525488885812910803106833635, 4.45618174639019371561735875075, 4.67745359171453829973301412698, 5.16514684409127151400237530347, 5.80039929502246160298895306554, 5.93101537184387399017731578862, 6.46383618078303211486933142084, 7.23575387616021305799087228683, 7.39857562650960081117665772677, 7.927858138553774775359398639272, 8.379895306690642539680548967067, 8.388318984386349664741614546375, 8.937800276168340691835853866026, 9.501418085284460779059325874929, 9.615517958435247271688481508354