| L(s) = 1 | + 8·4-s + 140·13-s + 56·19-s + 125·25-s + 308·31-s − 110·37-s − 1.04e3·43-s + 1.12e3·52-s + 182·61-s − 512·64-s + 880·67-s + 1.19e3·73-s + 448·76-s − 884·79-s + 2.66e3·97-s + 1.00e3·100-s + 1.82e3·103-s + 646·109-s + 1.33e3·121-s + 2.46e3·124-s + 127-s + 131-s + 137-s + 139-s − 880·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4-s + 2.98·13-s + 0.676·19-s + 25-s + 1.78·31-s − 0.488·37-s − 3.68·43-s + 2.98·52-s + 0.382·61-s − 64-s + 1.60·67-s + 1.90·73-s + 0.676·76-s − 1.25·79-s + 2.78·97-s + 100-s + 1.74·103-s + 0.567·109-s + 121-s + 1.78·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.488·148-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.909999406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.909999406\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 163 T + p^{3} T^{2} )( 1 + 107 T + p^{3} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 289 T + p^{3} T^{2} )( 1 - 19 T + p^{3} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 323 T + p^{3} T^{2} )( 1 + 433 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 520 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 901 T + p^{3} T^{2} )( 1 + 719 T + p^{3} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 1007 T + p^{3} T^{2} )( 1 + 127 T + p^{3} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 919 T + p^{3} T^{2} )( 1 - 271 T + p^{3} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 503 T + p^{3} T^{2} )( 1 + 1387 T + p^{3} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1330 T + p^{3} T^{2} )^{2} \) |
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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
−11.12925556928634260238968211254, −10.52443727281703733767782848486, −10.09418215295189394982834103291, −9.793669872456783475900362390348, −8.767380475425838888391947251652, −8.734851714721924688401780240230, −8.318220335954036810235810003952, −7.76970550240274935507859973537, −7.09584968711626688931932193397, −6.49770407250731333564637511219, −6.44695718963114991338191185798, −5.98155052175071680745929254055, −5.11591640376003062515235918775, −4.80332429495097302717263278124, −3.76920995117335966210653982614, −3.46495679072924515643198194158, −2.93106358904960128310417592302, −2.01047719789283763296679350191, −1.37395620063031139417672714403, −0.78048792638021808788462561266,
0.78048792638021808788462561266, 1.37395620063031139417672714403, 2.01047719789283763296679350191, 2.93106358904960128310417592302, 3.46495679072924515643198194158, 3.76920995117335966210653982614, 4.80332429495097302717263278124, 5.11591640376003062515235918775, 5.98155052175071680745929254055, 6.44695718963114991338191185798, 6.49770407250731333564637511219, 7.09584968711626688931932193397, 7.76970550240274935507859973537, 8.318220335954036810235810003952, 8.734851714721924688401780240230, 8.767380475425838888391947251652, 9.793669872456783475900362390348, 10.09418215295189394982834103291, 10.52443727281703733767782848486, 11.12925556928634260238968211254
