| L(s) = 1 | − 3·4-s + 5-s − 4·9-s − 6·11-s + 5·16-s − 2·19-s − 3·20-s + 25-s − 12·29-s + 6·31-s + 12·36-s − 12·41-s + 18·44-s − 4·45-s + 6·49-s − 6·55-s + 6·59-s + 16·61-s − 3·64-s − 12·71-s + 6·76-s − 10·79-s + 5·80-s + 7·81-s − 2·95-s + 24·99-s − 3·100-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 0.447·5-s − 4/3·9-s − 1.80·11-s + 5/4·16-s − 0.458·19-s − 0.670·20-s + 1/5·25-s − 2.22·29-s + 1.07·31-s + 2·36-s − 1.87·41-s + 2.71·44-s − 0.596·45-s + 6/7·49-s − 0.809·55-s + 0.781·59-s + 2.04·61-s − 3/8·64-s − 1.42·71-s + 0.688·76-s − 1.12·79-s + 0.559·80-s + 7/9·81-s − 0.205·95-s + 2.41·99-s − 0.299·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( 1 - T \) |
| 11 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
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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
−10.76901631957019374066241473658, −10.13605068220931389345708770409, −9.858589647788108281041476654884, −9.114317305715042420259307385768, −8.548925953485948244638238840664, −8.351830733471462569370341538818, −7.66787121601823913740640324076, −6.86292423042839496475885326716, −5.78938268791392875304163771661, −5.48080125027536949707953604299, −5.01827548961295327701762406430, −4.13903181362162610531995735363, −3.22585393988007919858454432865, −2.32936093136540293812935385391, 0,
2.32936093136540293812935385391, 3.22585393988007919858454432865, 4.13903181362162610531995735363, 5.01827548961295327701762406430, 5.48080125027536949707953604299, 5.78938268791392875304163771661, 6.86292423042839496475885326716, 7.66787121601823913740640324076, 8.351830733471462569370341538818, 8.548925953485948244638238840664, 9.114317305715042420259307385768, 9.858589647788108281041476654884, 10.13605068220931389345708770409, 10.76901631957019374066241473658
