| L(s) = 1 | + 7·4-s − 46·9-s − 60·11-s − 15·16-s − 208·19-s + 132·29-s + 388·31-s − 322·36-s − 252·41-s − 420·44-s + 202·49-s − 864·59-s − 1.22e3·61-s − 553·64-s − 348·71-s − 1.45e3·76-s − 796·79-s + 1.38e3·81-s − 1.26e3·89-s + 2.76e3·99-s − 2.46e3·101-s − 1.22e3·109-s + 924·116-s + 38·121-s + 2.71e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 7/8·4-s − 1.70·9-s − 1.64·11-s − 0.234·16-s − 2.51·19-s + 0.845·29-s + 2.24·31-s − 1.49·36-s − 0.959·41-s − 1.43·44-s + 0.588·49-s − 1.90·59-s − 2.56·61-s − 1.08·64-s − 0.581·71-s − 2.19·76-s − 1.13·79-s + 1.90·81-s − 1.50·89-s + 2.80·99-s − 2.42·101-s − 1.07·109-s + 0.739·116-s + 0.0285·121-s + 1.96·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 46 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 202 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2278 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 104 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 22570 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 66 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 194 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58870 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 126 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 8470 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 83954 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 291670 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 432 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 p T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 117578 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 174 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 646990 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 398 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 457990 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 630 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 382850 T^{2} + p^{6} T^{4} \) |
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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
−10.61406195755511322340145654864, −10.41243398525853930244559124135, −9.717582541826570954763122946192, −9.002504410272909045244761497841, −8.605255457985561724442613959472, −8.096343792674737553908227858991, −8.084760451494656375176077203794, −7.24903915941787346754182018648, −6.66260908414996070240397985392, −6.12300832119409654756342597977, −6.06553492985283713231707035219, −5.25095487018032995576094575912, −4.66357555434667859352350636077, −4.26135670094577363948482707570, −3.08168957013147942596082013563, −2.67804981185690727043470448689, −2.49262090035755248090166009305, −1.53445848275965997101231738413, 0, 0,
1.53445848275965997101231738413, 2.49262090035755248090166009305, 2.67804981185690727043470448689, 3.08168957013147942596082013563, 4.26135670094577363948482707570, 4.66357555434667859352350636077, 5.25095487018032995576094575912, 6.06553492985283713231707035219, 6.12300832119409654756342597977, 6.66260908414996070240397985392, 7.24903915941787346754182018648, 8.084760451494656375176077203794, 8.096343792674737553908227858991, 8.605255457985561724442613959472, 9.002504410272909045244761497841, 9.717582541826570954763122946192, 10.41243398525853930244559124135, 10.61406195755511322340145654864
