| L(s) = 1 | + 2-s + 3-s + 5-s + 6-s − 8-s + 10-s + 4·11-s − 4·13-s + 15-s − 16-s + 6·17-s + 4·22-s + 8·23-s − 24-s − 4·26-s − 27-s + 20·29-s + 30-s + 8·31-s + 4·33-s + 6·34-s − 2·37-s − 4·39-s − 40-s − 4·41-s + 16·43-s + 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 0.316·10-s + 1.20·11-s − 1.10·13-s + 0.258·15-s − 1/4·16-s + 1.45·17-s + 0.852·22-s + 1.66·23-s − 0.204·24-s − 0.784·26-s − 0.192·27-s + 3.71·29-s + 0.182·30-s + 1.43·31-s + 0.696·33-s + 1.02·34-s − 0.328·37-s − 0.640·39-s − 0.158·40-s − 0.624·41-s + 2.43·43-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.506873580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.506873580\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.736394053673218938829739406918, −9.339775061884020471535092823995, −8.773089099611289049369315836972, −8.648189804523516725023474145827, −8.208069100639170014478029536866, −7.64867778273108922106196192359, −7.18450498058194960604027898687, −6.92350029468128835494693937043, −6.25137104926623693209107340998, −6.13646854644971518825143971446, −5.60442239270420786798690458103, −4.82407209952283996064292501350, −4.60074184654846919093146647870, −4.56962705750483681018719945804, −3.42740777484311145525261574211, −3.32865706815750983037237151670, −2.68931538519026675552111457280, −2.42460416094508963198132706699, −1.25767150216837542259253513888, −0.969661224762805146998784920337,
0.969661224762805146998784920337, 1.25767150216837542259253513888, 2.42460416094508963198132706699, 2.68931538519026675552111457280, 3.32865706815750983037237151670, 3.42740777484311145525261574211, 4.56962705750483681018719945804, 4.60074184654846919093146647870, 4.82407209952283996064292501350, 5.60442239270420786798690458103, 6.13646854644971518825143971446, 6.25137104926623693209107340998, 6.92350029468128835494693937043, 7.18450498058194960604027898687, 7.64867778273108922106196192359, 8.208069100639170014478029536866, 8.648189804523516725023474145827, 8.773089099611289049369315836972, 9.339775061884020471535092823995, 9.736394053673218938829739406918
