| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s + 9-s − 2·10-s + 3·11-s + 12-s + 13-s + 2·15-s + 16-s − 17-s − 18-s + 3·19-s + 2·20-s − 3·22-s − 24-s − 25-s − 26-s + 27-s + 3·29-s − 2·30-s + 4·31-s − 32-s + 3·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.904·11-s + 0.288·12-s + 0.277·13-s + 0.516·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.688·19-s + 0.447·20-s − 0.639·22-s − 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.557·29-s − 0.365·30-s + 0.718·31-s − 0.176·32-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.313593202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.313593202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.618404057752291943367643428324, −7.88410064175701735824096266198, −7.10086085896993879645000800989, −6.36609787325588074799665492758, −5.76796744818104641814039616401, −4.67403452130842989355926942999, −3.68964912726581477742436961323, −2.76180092760599884329127348451, −1.88258517906705477543497928373, −1.01814568475901937041724011995,
1.01814568475901937041724011995, 1.88258517906705477543497928373, 2.76180092760599884329127348451, 3.68964912726581477742436961323, 4.67403452130842989355926942999, 5.76796744818104641814039616401, 6.36609787325588074799665492758, 7.10086085896993879645000800989, 7.88410064175701735824096266198, 8.618404057752291943367643428324
