L(s) = 1 | − 3·3-s − 4-s − 7·7-s + 6·9-s + 3·12-s − 4·13-s + 16-s + 19-s + 21·21-s − 2·25-s − 9·27-s + 7·28-s − 6·36-s − 3·37-s + 12·39-s + 43-s − 3·48-s + 23·49-s + 4·52-s − 3·57-s − 12·61-s − 42·63-s − 64-s + 2·67-s − 6·73-s + 6·75-s − 76-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 1/2·4-s − 2.64·7-s + 2·9-s + 0.866·12-s − 1.10·13-s + 1/4·16-s + 0.229·19-s + 4.58·21-s − 2/5·25-s − 1.73·27-s + 1.32·28-s − 36-s − 0.493·37-s + 1.92·39-s + 0.152·43-s − 0.433·48-s + 23/7·49-s + 0.554·52-s − 0.397·57-s − 1.53·61-s − 5.29·63-s − 1/8·64-s + 0.244·67-s − 0.702·73-s + 0.692·75-s − 0.114·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3708 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3708 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 103 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 108 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
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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
−12.45142127996617408126379938582, −11.90337927328674306525057081992, −11.10809491246833737433074112682, −10.33621232925751822764938822651, −9.923334500403239852609193874624, −9.598658298063641296575090737246, −8.910831833621615266926134800678, −7.57370742107072960804555749595, −6.95303205946709588301405919630, −6.36873393161333464228487463015, −5.84553951728082827322803021619, −5.07606574591929510242574115362, −4.11623608957528054560040047302, −3.07460668026695568250594068458, 0,
3.07460668026695568250594068458, 4.11623608957528054560040047302, 5.07606574591929510242574115362, 5.84553951728082827322803021619, 6.36873393161333464228487463015, 6.95303205946709588301405919630, 7.57370742107072960804555749595, 8.910831833621615266926134800678, 9.598658298063641296575090737246, 9.923334500403239852609193874624, 10.33621232925751822764938822651, 11.10809491246833737433074112682, 11.90337927328674306525057081992, 12.45142127996617408126379938582