| L(s) = 1 | − 2-s − 3·5-s − 7-s + 8-s + 3·10-s − 6·11-s − 2·13-s + 14-s − 16-s − 12·17-s − 14·19-s + 6·22-s + 3·23-s + 5·25-s + 2·26-s + 6·29-s − 2·31-s + 12·34-s + 3·35-s + 4·37-s + 14·38-s − 3·40-s − 2·43-s − 3·46-s − 5·50-s − 12·53-s + 18·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·5-s − 0.377·7-s + 0.353·8-s + 0.948·10-s − 1.80·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 2.91·17-s − 3.21·19-s + 1.27·22-s + 0.625·23-s + 25-s + 0.392·26-s + 1.11·29-s − 0.359·31-s + 2.05·34-s + 0.507·35-s + 0.657·37-s + 2.27·38-s − 0.474·40-s − 0.304·43-s − 0.442·46-s − 0.707·50-s − 1.64·53-s + 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.90450564140489019519298308146, −10.82172178761459126534247229491, −10.37963010064293986895324906345, −9.833794920243197303086819951413, −8.943575628321349450974308630359, −8.905452988034738469766559974726, −8.250330189856184009945970807373, −8.172850567326801726922966235812, −7.37744561624587485312104480999, −7.08339702267189722087157721419, −6.31597036463547930931719513459, −6.22272127375016884944758152832, −4.86655794452651834681308798853, −4.57197135055793564194285155868, −4.40168990874711540339157466811, −3.41515839077614286366190499300, −2.48335868333655857253174963387, −2.19438800668068560131758197913, 0, 0,
2.19438800668068560131758197913, 2.48335868333655857253174963387, 3.41515839077614286366190499300, 4.40168990874711540339157466811, 4.57197135055793564194285155868, 4.86655794452651834681308798853, 6.22272127375016884944758152832, 6.31597036463547930931719513459, 7.08339702267189722087157721419, 7.37744561624587485312104480999, 8.172850567326801726922966235812, 8.250330189856184009945970807373, 8.905452988034738469766559974726, 8.943575628321349450974308630359, 9.833794920243197303086819951413, 10.37963010064293986895324906345, 10.82172178761459126534247229491, 10.90450564140489019519298308146
