| L(s) = 1 | − 3·4-s − 10·5-s + 22·11-s − 7·16-s + 30·20-s + 75·25-s + 36·31-s − 66·44-s − 78·49-s − 220·55-s + 204·59-s + 69·64-s + 156·71-s + 70·80-s − 4·89-s − 225·100-s + 363·121-s − 108·124-s − 500·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 360·155-s + 157-s + ⋯ |
| L(s) = 1 | − 3/4·4-s − 2·5-s + 2·11-s − 0.437·16-s + 3/2·20-s + 3·25-s + 1.16·31-s − 3/2·44-s − 1.59·49-s − 4·55-s + 3.45·59-s + 1.07·64-s + 2.19·71-s + 7/8·80-s − 0.0449·89-s − 9/4·100-s + 3·121-s − 0.870·124-s − 4·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 2.32·155-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.184435535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.184435535\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 78 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 162 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 402 T^{2} + p^{4} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3522 T^{2} + p^{4} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 102 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 78 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 10638 T^{2} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13602 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−11.25946579749753796905904202857, −10.61414615836775017677238830271, −9.931119631226211048762280334786, −9.580161549977617725825212824531, −9.084163100437264029272454974741, −8.662474599469984518573554130797, −8.159345742428135514450714262116, −8.152450871173124757084201175585, −7.20968898667823856517125185813, −6.84795390293104492097642295673, −6.63300945157482431724516193468, −5.88659291936216063069131394204, −4.88280687063990011902003605870, −4.78709964565121954169614645242, −4.00857876958298732717803296327, −3.83898473998158960902208696932, −3.34477470609688061775631436523, −2.38696157012541458771915640738, −1.16057423029661443238809608699, −0.53589133734196648335068086781,
0.53589133734196648335068086781, 1.16057423029661443238809608699, 2.38696157012541458771915640738, 3.34477470609688061775631436523, 3.83898473998158960902208696932, 4.00857876958298732717803296327, 4.78709964565121954169614645242, 4.88280687063990011902003605870, 5.88659291936216063069131394204, 6.63300945157482431724516193468, 6.84795390293104492097642295673, 7.20968898667823856517125185813, 8.152450871173124757084201175585, 8.159345742428135514450714262116, 8.662474599469984518573554130797, 9.084163100437264029272454974741, 9.580161549977617725825212824531, 9.931119631226211048762280334786, 10.61414615836775017677238830271, 11.25946579749753796905904202857
