L(s) = 1 | − 4-s + 2·9-s + 2·11-s + 16-s − 8·19-s − 4·29-s − 20·31-s − 2·36-s − 2·44-s − 49-s − 20·59-s − 16·61-s − 64-s − 8·71-s + 8·76-s − 32·79-s − 5·81-s − 20·89-s + 4·99-s + 24·101-s + 28·109-s + 4·116-s + 3·121-s + 20·124-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 2/3·9-s + 0.603·11-s + 1/4·16-s − 1.83·19-s − 0.742·29-s − 3.59·31-s − 1/3·36-s − 0.301·44-s − 1/7·49-s − 2.60·59-s − 2.04·61-s − 1/8·64-s − 0.949·71-s + 0.917·76-s − 3.60·79-s − 5/9·81-s − 2.11·89-s + 0.402·99-s + 2.38·101-s + 2.68·109-s + 0.371·116-s + 3/11·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.509142714804071880169470379014, −7.68097621618888915672921080039, −7.62612423498325200928407991979, −7.30365362569966069266190589503, −6.91231864705942975771342471721, −6.42637692479441654839206374861, −5.92437302492834489911484717312, −5.86126551682800339783548104855, −5.35915877197077491957819450341, −4.70665090138765286883566571405, −4.48509324650060004897813511000, −4.11690814864027198690782704995, −3.77622558572754632179191100898, −3.24280447332641273002125455754, −2.87957907785677397850662379278, −1.93571575960425887653193828978, −1.77598099464084405532041865821, −1.33820896313636744418261307428, 0, 0,
1.33820896313636744418261307428, 1.77598099464084405532041865821, 1.93571575960425887653193828978, 2.87957907785677397850662379278, 3.24280447332641273002125455754, 3.77622558572754632179191100898, 4.11690814864027198690782704995, 4.48509324650060004897813511000, 4.70665090138765286883566571405, 5.35915877197077491957819450341, 5.86126551682800339783548104855, 5.92437302492834489911484717312, 6.42637692479441654839206374861, 6.91231864705942975771342471721, 7.30365362569966069266190589503, 7.62612423498325200928407991979, 7.68097621618888915672921080039, 8.509142714804071880169470379014