| L(s) = 1 | − 4-s − 9-s − 8·11-s + 16-s + 8·19-s − 4·29-s − 2·31-s + 36-s − 12·41-s + 8·44-s + 14·49-s + 8·59-s − 12·61-s − 64-s − 24·71-s − 8·76-s + 32·79-s + 81-s + 36·89-s + 8·99-s + 4·101-s − 12·109-s + 4·116-s + 26·121-s + 2·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 2.41·11-s + 1/4·16-s + 1.83·19-s − 0.742·29-s − 0.359·31-s + 1/6·36-s − 1.87·41-s + 1.20·44-s + 2·49-s + 1.04·59-s − 1.53·61-s − 1/8·64-s − 2.84·71-s − 0.917·76-s + 3.60·79-s + 1/9·81-s + 3.81·89-s + 0.804·99-s + 0.398·101-s − 1.14·109-s + 0.371·116-s + 2.36·121-s + 0.179·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8958634970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8958634970\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.439444449280731202547618255361, −7.899948464414646298245356712287, −7.88826676411114086015180268359, −7.50018854205871087950652031116, −7.23916183072290280641335708240, −6.75951130252152462706817307619, −6.19459738101779364208003183038, −5.78901527262405287273900161174, −5.47006550363578271270820123651, −5.22790709303042256515861386380, −4.75531691772364670746057878341, −4.74743975457673917610222720434, −3.68811900055568118011006169861, −3.65552314134243126551968418867, −3.13411649559799047268277994354, −2.65480422354029212824526194435, −2.30751588920000707623606163577, −1.71454351928578156580870618455, −0.956751570334089312185221692604, −0.30605994068446972458362172801,
0.30605994068446972458362172801, 0.956751570334089312185221692604, 1.71454351928578156580870618455, 2.30751588920000707623606163577, 2.65480422354029212824526194435, 3.13411649559799047268277994354, 3.65552314134243126551968418867, 3.68811900055568118011006169861, 4.74743975457673917610222720434, 4.75531691772364670746057878341, 5.22790709303042256515861386380, 5.47006550363578271270820123651, 5.78901527262405287273900161174, 6.19459738101779364208003183038, 6.75951130252152462706817307619, 7.23916183072290280641335708240, 7.50018854205871087950652031116, 7.88826676411114086015180268359, 7.899948464414646298245356712287, 8.439444449280731202547618255361
