| L(s) = 1 | + 2·2-s − 4·3-s + 3·4-s − 8·6-s + 4·8-s + 8·9-s + 4·11-s − 12·12-s + 4·13-s + 5·16-s − 8·17-s + 16·18-s + 4·19-s + 8·22-s + 8·23-s − 16·24-s + 8·26-s − 12·27-s − 4·29-s + 6·32-s − 16·33-s − 16·34-s + 24·36-s + 4·37-s + 8·38-s − 16·39-s + 8·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.30·3-s + 3/2·4-s − 3.26·6-s + 1.41·8-s + 8/3·9-s + 1.20·11-s − 3.46·12-s + 1.10·13-s + 5/4·16-s − 1.94·17-s + 3.77·18-s + 0.917·19-s + 1.70·22-s + 1.66·23-s − 3.26·24-s + 1.56·26-s − 2.30·27-s − 0.742·29-s + 1.06·32-s − 2.78·33-s − 2.74·34-s + 4·36-s + 0.657·37-s + 1.29·38-s − 2.56·39-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.766302881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.766302881\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 20 T + 216 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 208 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 166 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 152 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−9.086800787097755157525184194109, −9.003555614241187848308917353347, −8.201254523175589357966630569448, −7.83604784604750215321483365415, −7.05090813215107186176798803581, −7.00895855375079864722618698888, −6.63541974228189732737278139590, −6.31109121075967209934194306496, −5.84386304574593071301152184994, −5.77550573270655507526439868050, −5.11804160807057061050623341678, −4.99364335565615103537599241554, −4.35178174253138749067332625001, −4.20516015212238769926970896368, −3.59254636969922239947644205020, −3.26874178011577849744652129922, −2.34212565232817952324516023042, −1.93025434238853365646047032315, −0.962890402474903067441645072620, −0.78851705512059124926063937719,
0.78851705512059124926063937719, 0.962890402474903067441645072620, 1.93025434238853365646047032315, 2.34212565232817952324516023042, 3.26874178011577849744652129922, 3.59254636969922239947644205020, 4.20516015212238769926970896368, 4.35178174253138749067332625001, 4.99364335565615103537599241554, 5.11804160807057061050623341678, 5.77550573270655507526439868050, 5.84386304574593071301152184994, 6.31109121075967209934194306496, 6.63541974228189732737278139590, 7.00895855375079864722618698888, 7.05090813215107186176798803581, 7.83604784604750215321483365415, 8.201254523175589357966630569448, 9.003555614241187848308917353347, 9.086800787097755157525184194109
