| L(s) = 1 | − 4·3-s + 2·5-s − 4·7-s + 8·9-s − 2·13-s − 8·15-s − 4·17-s − 8·19-s + 16·21-s − 4·23-s + 3·25-s − 12·27-s − 8·29-s − 8·35-s − 8·37-s + 8·39-s + 4·41-s − 12·43-s + 16·45-s + 4·47-s − 2·49-s + 16·51-s − 12·53-s + 32·57-s − 8·59-s + 8·61-s − 32·63-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.894·5-s − 1.51·7-s + 8/3·9-s − 0.554·13-s − 2.06·15-s − 0.970·17-s − 1.83·19-s + 3.49·21-s − 0.834·23-s + 3/5·25-s − 2.30·27-s − 1.48·29-s − 1.35·35-s − 1.31·37-s + 1.28·39-s + 0.624·41-s − 1.82·43-s + 2.38·45-s + 0.583·47-s − 2/7·49-s + 2.24·51-s − 1.64·53-s + 4.23·57-s − 1.04·59-s + 1.02·61-s − 4.03·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 104 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 10 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 166 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−10.76892515625415663970430565533, −10.25584773966495624700114241241, −9.829230974950063319408995318639, −9.678936856600928625873228805734, −9.012565739669430604925793247991, −8.598884687501890813542150317531, −7.80522255468531574216016289324, −7.14545383899240331564458120240, −6.55526465178781342584022025280, −6.51325824083471360453124345990, −6.00587786061643390440460691024, −5.78411318658370835184766096339, −4.95608605869718238568819305142, −4.83753720367894891817375253391, −3.96423568809408738269943956599, −3.38062496686782974922466034617, −2.30123926700117947836741088811, −1.74096157805781498233793892100, 0, 0,
1.74096157805781498233793892100, 2.30123926700117947836741088811, 3.38062496686782974922466034617, 3.96423568809408738269943956599, 4.83753720367894891817375253391, 4.95608605869718238568819305142, 5.78411318658370835184766096339, 6.00587786061643390440460691024, 6.51325824083471360453124345990, 6.55526465178781342584022025280, 7.14545383899240331564458120240, 7.80522255468531574216016289324, 8.598884687501890813542150317531, 9.012565739669430604925793247991, 9.678936856600928625873228805734, 9.829230974950063319408995318639, 10.25584773966495624700114241241, 10.76892515625415663970430565533
