| L(s) = 1 | − 2-s − 4·7-s + 8-s + 11-s − 8·13-s + 4·14-s − 16-s + 3·17-s + 19-s − 22-s − 3·23-s + 5·25-s + 8·26-s + 18·29-s − 2·31-s − 3·34-s + 7·37-s − 38-s + 12·41-s + 22·43-s + 3·46-s − 3·47-s + 9·49-s − 5·50-s − 4·56-s − 18·58-s + 9·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.51·7-s + 0.353·8-s + 0.301·11-s − 2.21·13-s + 1.06·14-s − 1/4·16-s + 0.727·17-s + 0.229·19-s − 0.213·22-s − 0.625·23-s + 25-s + 1.56·26-s + 3.34·29-s − 0.359·31-s − 0.514·34-s + 1.15·37-s − 0.162·38-s + 1.87·41-s + 3.35·43-s + 0.442·46-s − 0.437·47-s + 9/7·49-s − 0.707·50-s − 0.534·56-s − 2.36·58-s + 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.210779363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.210779363\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−9.652948014956762117777238651995, −9.585008766208023078537445409950, −9.032450223150904502466295136007, −8.762017923723495760083096159468, −8.068247593891087323826028353156, −7.81468494840802803262887427405, −7.36183957453432619304249379309, −7.03823068505168467452240858567, −6.53821597116820243167318879751, −6.24288980124215225267153106198, −5.71654506427053333093812376522, −5.17090620876012841715518721636, −4.68973275864814302578313536073, −4.24670786852414636716958178473, −3.78214763964455677313804430139, −2.96060924653939199280351827165, −2.52303483453212598594153913163, −2.44406063584878451689783852979, −0.901863701225170832255194409845, −0.70268164688304263305963699137,
0.70268164688304263305963699137, 0.901863701225170832255194409845, 2.44406063584878451689783852979, 2.52303483453212598594153913163, 2.96060924653939199280351827165, 3.78214763964455677313804430139, 4.24670786852414636716958178473, 4.68973275864814302578313536073, 5.17090620876012841715518721636, 5.71654506427053333093812376522, 6.24288980124215225267153106198, 6.53821597116820243167318879751, 7.03823068505168467452240858567, 7.36183957453432619304249379309, 7.81468494840802803262887427405, 8.068247593891087323826028353156, 8.762017923723495760083096159468, 9.032450223150904502466295136007, 9.585008766208023078537445409950, 9.652948014956762117777238651995
