| L(s) = 1 | − 4·7-s + 6·9-s + 12·17-s + 16·23-s + 6·25-s − 20·31-s + 12·41-s + 4·47-s − 2·49-s − 24·63-s + 20·71-s − 4·73-s + 8·79-s + 27·81-s + 12·89-s + 4·97-s − 16·103-s + 28·113-s − 48·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 72·153-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 2·9-s + 2.91·17-s + 3.33·23-s + 6/5·25-s − 3.59·31-s + 1.87·41-s + 0.583·47-s − 2/7·49-s − 3.02·63-s + 2.37·71-s − 0.468·73-s + 0.900·79-s + 3·81-s + 1.27·89-s + 0.406·97-s − 1.57·103-s + 2.63·113-s − 4.40·119-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 5.82·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.926484220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.926484220\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.090322850212685434003947732148, −8.501790641516419792698386485765, −7.84246254566311702676034805965, −7.57280546991219335579010534867, −7.19473966854861786710649131782, −7.10041375439695171027985830439, −6.75804817318721114003935994198, −6.24213511517197549276612964489, −5.74790785468569100893343031948, −5.41428232339832053708538941923, −4.89100168812448932038231351354, −4.83534112165499395539914685161, −3.92811730028493936210430088391, −3.64429077591181687790501944794, −3.22506366195855257825503171715, −3.14981636149932821211840835164, −2.32121538376297524152832952161, −1.61858460732684329621560685007, −0.978048321201144394584188386267, −0.810237419234173064450958648354,
0.810237419234173064450958648354, 0.978048321201144394584188386267, 1.61858460732684329621560685007, 2.32121538376297524152832952161, 3.14981636149932821211840835164, 3.22506366195855257825503171715, 3.64429077591181687790501944794, 3.92811730028493936210430088391, 4.83534112165499395539914685161, 4.89100168812448932038231351354, 5.41428232339832053708538941923, 5.74790785468569100893343031948, 6.24213511517197549276612964489, 6.75804817318721114003935994198, 7.10041375439695171027985830439, 7.19473966854861786710649131782, 7.57280546991219335579010534867, 7.84246254566311702676034805965, 8.501790641516419792698386485765, 9.090322850212685434003947732148
