| L(s) = 1 | − 2·4-s + 7-s + 9·19-s + 5·25-s − 2·28-s − 3·31-s − 10·37-s + 26·43-s − 6·49-s − 15·61-s + 8·64-s + 16·67-s + 27·73-s − 18·76-s + 4·79-s − 10·100-s + 6·103-s − 19·109-s − 11·121-s + 6·124-s + 127-s + 131-s + 9·133-s + 137-s + 139-s + 20·148-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s + 0.377·7-s + 2.06·19-s + 25-s − 0.377·28-s − 0.538·31-s − 1.64·37-s + 3.96·43-s − 6/7·49-s − 1.92·61-s + 64-s + 1.95·67-s + 3.16·73-s − 2.06·76-s + 0.450·79-s − 100-s + 0.591·103-s − 1.81·109-s − 121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.780·133-s + 0.0854·137-s + 0.0848·139-s + 1.64·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.068089793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.068089793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−12.55212682660438983472157645227, −12.54713968965605036693842573810, −11.97555252763756146134648577293, −11.08995198316239262165038813214, −11.05401100186968864581132446171, −10.38152969226145926335818366762, −9.602976770644561260497905342236, −9.288465943745865816007739079544, −9.036360803211178457930941565742, −8.270771669586352006171892199710, −7.76823041981533334697404465071, −7.27141302921257274556978944484, −6.65677308889071393553363619996, −5.80736724567364436227539884582, −5.14979491245933178669563729918, −4.91529950925449047407897129273, −4.00740754937879545210542399250, −3.42704674854982646766031179707, −2.43514023894947911090584523261, −1.03997947285202387552651717274,
1.03997947285202387552651717274, 2.43514023894947911090584523261, 3.42704674854982646766031179707, 4.00740754937879545210542399250, 4.91529950925449047407897129273, 5.14979491245933178669563729918, 5.80736724567364436227539884582, 6.65677308889071393553363619996, 7.27141302921257274556978944484, 7.76823041981533334697404465071, 8.270771669586352006171892199710, 9.036360803211178457930941565742, 9.288465943745865816007739079544, 9.602976770644561260497905342236, 10.38152969226145926335818366762, 11.05401100186968864581132446171, 11.08995198316239262165038813214, 11.97555252763756146134648577293, 12.54713968965605036693842573810, 12.55212682660438983472157645227
