| L(s) = 1 | + 2·3-s − 2·5-s + 6·7-s + 3·9-s + 2·11-s + 6·13-s − 4·15-s + 2·19-s + 12·21-s − 4·23-s + 3·25-s + 4·27-s + 6·29-s + 4·33-s − 12·35-s + 2·37-s + 12·39-s − 6·41-s + 6·43-s − 6·45-s − 4·47-s + 18·49-s + 8·53-s − 4·55-s + 4·57-s − 12·59-s + 18·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 2.26·7-s + 9-s + 0.603·11-s + 1.66·13-s − 1.03·15-s + 0.458·19-s + 2.61·21-s − 0.834·23-s + 3/5·25-s + 0.769·27-s + 1.11·29-s + 0.696·33-s − 2.02·35-s + 0.328·37-s + 1.92·39-s − 0.937·41-s + 0.914·43-s − 0.894·45-s − 0.583·47-s + 18/7·49-s + 1.09·53-s − 0.539·55-s + 0.529·57-s − 1.56·59-s + 2.26·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5198400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5198400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.426417170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.426417170\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 158 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 238 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.813835114355619889166417595921, −8.744196797636863967857935452811, −8.423728158455591085787733275322, −8.124133534546338764572201047299, −7.71505825470713131652386875527, −7.65457525873720904458324422917, −7.03025436768784851638368602193, −6.53555781010791228863114693826, −6.21352430822398207760285743444, −5.60110665967847373077520919781, −5.02787244041845602740975144412, −4.76791339661793130876811602081, −4.16990789957791228826056429269, −4.07552928210134086867168625705, −3.36818905496675062328644755452, −3.26000788720588408447623475329, −2.15554182042046854130630956339, −2.08156270990990432656257787061, −1.15281659063469294481992806686, −1.03759875975400455200289852186,
1.03759875975400455200289852186, 1.15281659063469294481992806686, 2.08156270990990432656257787061, 2.15554182042046854130630956339, 3.26000788720588408447623475329, 3.36818905496675062328644755452, 4.07552928210134086867168625705, 4.16990789957791228826056429269, 4.76791339661793130876811602081, 5.02787244041845602740975144412, 5.60110665967847373077520919781, 6.21352430822398207760285743444, 6.53555781010791228863114693826, 7.03025436768784851638368602193, 7.65457525873720904458324422917, 7.71505825470713131652386875527, 8.124133534546338764572201047299, 8.423728158455591085787733275322, 8.744196797636863967857935452811, 8.813835114355619889166417595921
