| L(s) = 1 | + 3-s − 5-s + 7-s − 9-s + 2·11-s + 4·13-s − 15-s − 3·19-s + 21-s + 4·23-s + 25-s + 29-s + 2·31-s + 2·33-s − 35-s − 2·37-s + 4·39-s − 7·41-s − 10·43-s + 45-s + 6·47-s − 5·49-s + 7·53-s − 2·55-s − 3·57-s − 11·59-s − 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s − 1/3·9-s + 0.603·11-s + 1.10·13-s − 0.258·15-s − 0.688·19-s + 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.185·29-s + 0.359·31-s + 0.348·33-s − 0.169·35-s − 0.328·37-s + 0.640·39-s − 1.09·41-s − 1.52·43-s + 0.149·45-s + 0.875·47-s − 5/7·49-s + 0.961·53-s − 0.269·55-s − 0.397·57-s − 1.43·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.277315596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.277315596\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 12 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 72 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 94 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 9 T + 78 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−15.9373995436, −15.4978513196, −15.1258359644, −14.6709890208, −14.1435661723, −13.7540923111, −13.2500062080, −12.7342832127, −12.0752932361, −11.6034643339, −11.2121058452, −10.5749337883, −10.1635187279, −9.25040281270, −8.80993517265, −8.42592391600, −7.98339292582, −7.12769501227, −6.60130868809, −5.95902663549, −5.06997635456, −4.35708721587, −3.57679475691, −2.90071851112, −1.56053455820,
1.56053455820, 2.90071851112, 3.57679475691, 4.35708721587, 5.06997635456, 5.95902663549, 6.60130868809, 7.12769501227, 7.98339292582, 8.42592391600, 8.80993517265, 9.25040281270, 10.1635187279, 10.5749337883, 11.2121058452, 11.6034643339, 12.0752932361, 12.7342832127, 13.2500062080, 13.7540923111, 14.1435661723, 14.6709890208, 15.1258359644, 15.4978513196, 15.9373995436
