| L(s) = 1 | − 3-s + 3·5-s − 2·9-s + 2·11-s − 3·15-s + 10·23-s + 4·25-s + 5·27-s − 31-s − 2·33-s + 6·37-s − 6·45-s + 4·47-s + 8·49-s − 7·53-s + 6·55-s + 7·67-s − 10·69-s − 71-s − 4·75-s + 81-s + 5·89-s + 93-s − 4·97-s − 4·99-s − 11·103-s − 6·111-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 2/3·9-s + 0.603·11-s − 0.774·15-s + 2.08·23-s + 4/5·25-s + 0.962·27-s − 0.179·31-s − 0.348·33-s + 0.986·37-s − 0.894·45-s + 0.583·47-s + 8/7·49-s − 0.961·53-s + 0.809·55-s + 0.855·67-s − 1.20·69-s − 0.118·71-s − 0.461·75-s + 1/9·81-s + 0.529·89-s + 0.103·93-s − 0.406·97-s − 0.402·99-s − 1.08·103-s − 0.569·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.488072036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.488072036\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 39 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−7.81490956631735883196439282134, −7.24778325597194871990959210795, −6.83181829297048305928022940315, −6.50261056136448259948151625674, −6.07111342419992463624540374903, −5.66782266903990255049563215136, −5.34227182835164058343884335647, −4.90020439549167858036084805778, −4.42851366742501083311110067889, −3.77192042006366392557449347373, −3.08868726901837514029882940909, −2.69784764105751400513264512444, −2.11645839465009846006285442470, −1.33794155319946546676789167247, −0.74130238237165472085202668292,
0.74130238237165472085202668292, 1.33794155319946546676789167247, 2.11645839465009846006285442470, 2.69784764105751400513264512444, 3.08868726901837514029882940909, 3.77192042006366392557449347373, 4.42851366742501083311110067889, 4.90020439549167858036084805778, 5.34227182835164058343884335647, 5.66782266903990255049563215136, 6.07111342419992463624540374903, 6.50261056136448259948151625674, 6.83181829297048305928022940315, 7.24778325597194871990959210795, 7.81490956631735883196439282134
