| L(s) = 1 | + 2·2-s + 3·4-s − 7-s + 4·8-s + 6·11-s − 5·13-s − 2·14-s + 5·16-s + 6·17-s + 4·19-s + 12·22-s + 6·23-s + 5·25-s − 10·26-s − 3·28-s + 6·29-s − 2·31-s + 6·32-s + 12·34-s + 37-s + 8·38-s − 6·41-s + 43-s + 18·44-s + 12·46-s + 12·47-s − 6·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.377·7-s + 1.41·8-s + 1.80·11-s − 1.38·13-s − 0.534·14-s + 5/4·16-s + 1.45·17-s + 0.917·19-s + 2.55·22-s + 1.25·23-s + 25-s − 1.96·26-s − 0.566·28-s + 1.11·29-s − 0.359·31-s + 1.06·32-s + 2.05·34-s + 0.164·37-s + 1.29·38-s − 0.937·41-s + 0.152·43-s + 2.71·44-s + 1.76·46-s + 1.75·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.828976378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.828976378\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−10.07410273012661055759578126232, −9.834200577861963653558935819176, −9.079530922070143779683080894501, −8.959865939426136703937909115793, −8.436941407549846736404055352110, −7.53110057991743244949354691771, −7.40621920555462411158286841358, −7.15235118997075856624656702725, −6.44416954844267361020173181362, −6.35293121075353093682762361401, −5.69751939932530624951526541798, −5.24357758870821828163612566850, −4.76574011115947488832893584227, −4.58314249904603549018329730152, −3.61524966323110508303152505132, −3.61238424405171559155945006030, −2.84334288646061969186461333611, −2.58924677193309664956985900145, −1.45685937729851190905882712233, −1.05265602894310832411085821794,
1.05265602894310832411085821794, 1.45685937729851190905882712233, 2.58924677193309664956985900145, 2.84334288646061969186461333611, 3.61238424405171559155945006030, 3.61524966323110508303152505132, 4.58314249904603549018329730152, 4.76574011115947488832893584227, 5.24357758870821828163612566850, 5.69751939932530624951526541798, 6.35293121075353093682762361401, 6.44416954844267361020173181362, 7.15235118997075856624656702725, 7.40621920555462411158286841358, 7.53110057991743244949354691771, 8.436941407549846736404055352110, 8.959865939426136703937909115793, 9.079530922070143779683080894501, 9.834200577861963653558935819176, 10.07410273012661055759578126232
