| L(s) = 1 | + 3·3-s − 2·4-s − 7-s + 6·9-s − 6·12-s − 9·19-s − 3·21-s + 9·27-s + 2·28-s + 15·31-s − 12·36-s + 37-s + 10·43-s − 6·49-s − 27·57-s + 12·61-s − 6·63-s + 8·64-s + 11·67-s + 27·73-s + 18·76-s + 13·79-s + 9·81-s + 6·84-s + 45·93-s − 33·103-s − 18·108-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 4-s − 0.377·7-s + 2·9-s − 1.73·12-s − 2.06·19-s − 0.654·21-s + 1.73·27-s + 0.377·28-s + 2.69·31-s − 2·36-s + 0.164·37-s + 1.52·43-s − 6/7·49-s − 3.57·57-s + 1.53·61-s − 0.755·63-s + 64-s + 1.34·67-s + 3.16·73-s + 2.06·76-s + 1.46·79-s + 81-s + 0.654·84-s + 4.66·93-s − 3.25·103-s − 1.73·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.345080248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.345080248\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−11.07819973987764280653510974994, −10.37650257749575262375681195705, −9.999621213421722491316626640805, −9.641709154406238161222607458249, −9.264492018743605027303909701158, −8.866237202378373240600551070480, −8.369414729250721848152321888788, −8.063727837778368209563984116739, −7.952681727012636919654588356996, −6.87867987815446069435897347762, −6.70720853015046547331354729237, −6.20240937006674184708994844485, −5.32965068511905031149613030409, −4.64585070953819687191436353598, −4.30830464324035628712094546409, −3.86116311364689938946201580604, −3.28687116431095061510459503434, −2.36153412573112177706269710039, −2.28147068853533253333925174862, −0.848079794222887665276495082755,
0.848079794222887665276495082755, 2.28147068853533253333925174862, 2.36153412573112177706269710039, 3.28687116431095061510459503434, 3.86116311364689938946201580604, 4.30830464324035628712094546409, 4.64585070953819687191436353598, 5.32965068511905031149613030409, 6.20240937006674184708994844485, 6.70720853015046547331354729237, 6.87867987815446069435897347762, 7.952681727012636919654588356996, 8.063727837778368209563984116739, 8.369414729250721848152321888788, 8.866237202378373240600551070480, 9.264492018743605027303909701158, 9.641709154406238161222607458249, 9.999621213421722491316626640805, 10.37650257749575262375681195705, 11.07819973987764280653510974994
