L(s) = 1 | + 2·3-s − 2·5-s + 7-s + 3·9-s + 5·11-s − 2·13-s − 4·15-s + 5·17-s + 4·19-s + 2·21-s − 7·23-s + 3·25-s + 4·27-s + 6·29-s + 6·31-s + 10·33-s − 2·35-s − 7·37-s − 4·39-s + 15·41-s − 8·43-s − 6·45-s − 10·47-s − 3·49-s + 10·51-s + 15·53-s − 10·55-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.377·7-s + 9-s + 1.50·11-s − 0.554·13-s − 1.03·15-s + 1.21·17-s + 0.917·19-s + 0.436·21-s − 1.45·23-s + 3/5·25-s + 0.769·27-s + 1.11·29-s + 1.07·31-s + 1.74·33-s − 0.338·35-s − 1.15·37-s − 0.640·39-s + 2.34·41-s − 1.21·43-s − 0.894·45-s − 1.45·47-s − 3/7·49-s + 1.40·51-s + 2.06·53-s − 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.974916833\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.974916833\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 7 | $D_{4}$ | \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 76 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 15 T + 128 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 15 T + 152 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T + 112 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - T - 114 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 150 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 3 T + 88 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 17 T + 174 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−8.601358538160058229187367823700, −8.513012574107738851453777616718, −8.125728187463283125375758241734, −7.921404438105427766392533749462, −7.43836417435185345354999384011, −7.16004176464981220129891423597, −6.59100561894644017456618011091, −6.58306695307714729680760656198, −5.73244919259082633904324530050, −5.52078459065643707212748824791, −4.73006779511847606493815703997, −4.64624501440403093630306086181, −3.91022293250671361286015060771, −3.89791818750877320500180721496, −3.25079129669448397673607378349, −3.03717918648948658688971072550, −2.32221787406482267844107067398, −1.85904083970071039005963747732, −1.16136551100561639049000530401, −0.74182929499742826857730364765,
0.74182929499742826857730364765, 1.16136551100561639049000530401, 1.85904083970071039005963747732, 2.32221787406482267844107067398, 3.03717918648948658688971072550, 3.25079129669448397673607378349, 3.89791818750877320500180721496, 3.91022293250671361286015060771, 4.64624501440403093630306086181, 4.73006779511847606493815703997, 5.52078459065643707212748824791, 5.73244919259082633904324530050, 6.58306695307714729680760656198, 6.59100561894644017456618011091, 7.16004176464981220129891423597, 7.43836417435185345354999384011, 7.921404438105427766392533749462, 8.125728187463283125375758241734, 8.513012574107738851453777616718, 8.601358538160058229187367823700