| L(s) = 1 | + 14·7-s − 34·13-s + 22·19-s − 116·31-s + 14·37-s − 52·43-s + 49·49-s − 2·61-s − 226·67-s + 86·73-s − 98·79-s − 476·91-s − 226·97-s − 34·103-s + 316·109-s + 62·121-s + 127-s + 131-s + 308·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 529·169-s + ⋯ |
| L(s) = 1 | + 2·7-s − 2.61·13-s + 1.15·19-s − 3.74·31-s + 0.378·37-s − 1.20·43-s + 49-s − 0.0327·61-s − 3.37·67-s + 1.17·73-s − 1.24·79-s − 5.23·91-s − 2.32·97-s − 0.330·103-s + 2.89·109-s + 0.512·121-s + 0.00787·127-s + 0.00763·131-s + 2.31·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3253085415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3253085415\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 62 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 142 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 562 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 62 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 58 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 1138 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 3998 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6242 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 113 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 5582 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 43 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 49 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12158 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15122 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 113 T + p^{2} T^{2} )^{2} \) |
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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
−9.151322178993067526984288735093, −8.318121973293312706216252529102, −8.066511388683606540014471879570, −7.56658495570221240184994912554, −7.47172580779290328455142438426, −7.01662014103510264702389824312, −6.96545665165691384801862194363, −5.83703262637351096249975713003, −5.76695676484671995615119351334, −5.17488583411634837167253108794, −5.02576195783100008192373951783, −4.59308931400440555740870924873, −4.35718440536272925596858803068, −3.54589940608350636810747312780, −3.22820327335984394095717492577, −2.54044068461591258692336215320, −2.08291220342908919062532842779, −1.66585891248269302550360882605, −1.27347129428514195813543059172, −0.12605478330273678697946599349,
0.12605478330273678697946599349, 1.27347129428514195813543059172, 1.66585891248269302550360882605, 2.08291220342908919062532842779, 2.54044068461591258692336215320, 3.22820327335984394095717492577, 3.54589940608350636810747312780, 4.35718440536272925596858803068, 4.59308931400440555740870924873, 5.02576195783100008192373951783, 5.17488583411634837167253108794, 5.76695676484671995615119351334, 5.83703262637351096249975713003, 6.96545665165691384801862194363, 7.01662014103510264702389824312, 7.47172580779290328455142438426, 7.56658495570221240184994912554, 8.066511388683606540014471879570, 8.318121973293312706216252529102, 9.151322178993067526984288735093
