| L(s) = 1 | − 2-s − 3·3-s + 4-s − 4·5-s + 3·6-s − 6·7-s − 8-s + 4·9-s + 4·10-s − 4·11-s − 3·12-s + 6·14-s + 12·15-s + 16-s − 7·17-s − 4·18-s − 4·20-s + 18·21-s + 4·22-s − 2·23-s + 3·24-s + 7·25-s − 6·28-s + 3·29-s − 12·30-s − 2·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.78·5-s + 1.22·6-s − 2.26·7-s − 0.353·8-s + 4/3·9-s + 1.26·10-s − 1.20·11-s − 0.866·12-s + 1.60·14-s + 3.09·15-s + 1/4·16-s − 1.69·17-s − 0.942·18-s − 0.894·20-s + 3.92·21-s + 0.852·22-s − 0.417·23-s + 0.612·24-s + 7/5·25-s − 1.13·28-s + 0.557·29-s − 2.19·30-s − 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 38 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 40 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 63 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 56 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 93 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T - 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 167 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T - 59 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
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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
−16.2104140539, −15.9481133064, −15.7730035457, −15.2912201377, −14.7959832518, −13.6349942861, −13.0613697059, −12.8317781765, −12.3444976943, −11.7835822640, −11.3318308393, −11.1375869673, −10.4141194183, −9.99385295629, −9.58331832105, −8.54164377410, −8.33475755656, −7.40347490100, −6.90393975972, −6.48831267428, −6.02422031231, −5.12774303045, −4.40602572808, −3.53010137455, −2.77936251566, 0, 0,
2.77936251566, 3.53010137455, 4.40602572808, 5.12774303045, 6.02422031231, 6.48831267428, 6.90393975972, 7.40347490100, 8.33475755656, 8.54164377410, 9.58331832105, 9.99385295629, 10.4141194183, 11.1375869673, 11.3318308393, 11.7835822640, 12.3444976943, 12.8317781765, 13.0613697059, 13.6349942861, 14.7959832518, 15.2912201377, 15.7730035457, 15.9481133064, 16.2104140539
