| L(s) = 1 | − 3·3-s + 2·5-s − 3·7-s + 4·9-s + 3·11-s + 13-s − 6·15-s + 11·17-s + 5·19-s + 9·21-s + 2·23-s + 3·25-s − 6·27-s − 10·29-s − 5·31-s − 9·33-s − 6·35-s + 4·37-s − 3·39-s − 13·41-s − 8·43-s + 8·45-s − 14·47-s − 4·49-s − 33·51-s + 6·55-s − 15·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s − 1.13·7-s + 4/3·9-s + 0.904·11-s + 0.277·13-s − 1.54·15-s + 2.66·17-s + 1.14·19-s + 1.96·21-s + 0.417·23-s + 3/5·25-s − 1.15·27-s − 1.85·29-s − 0.898·31-s − 1.56·33-s − 1.01·35-s + 0.657·37-s − 0.480·39-s − 2.03·41-s − 1.21·43-s + 1.19·45-s − 2.04·47-s − 4/7·49-s − 4.62·51-s + 0.809·55-s − 1.98·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 21 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 11 T + 61 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 15 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 65 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 13 T + 121 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 130 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 119 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7 T + 151 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 226 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T - 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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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
−7.72461929978765157566273487860, −7.22266349263049605087810173676, −6.73292998674861013119023952939, −6.67935056157105064923058187076, −6.26266052959310795509096411371, −6.00932208788318772113606105425, −5.56827772788063819312214295772, −5.38409653934961654966137334031, −5.01387005247890360020623258373, −5.00632052792353410505653707057, −4.05146868372649466566531579941, −3.54080698225297808582796089022, −3.44592046486286322546217075548, −3.21428846901977934002638162414, −2.48152779133608152816852651669, −1.68641615820823444187673095497, −1.28398725105889730217362062664, −1.22927341271740845674720379465, 0, 0,
1.22927341271740845674720379465, 1.28398725105889730217362062664, 1.68641615820823444187673095497, 2.48152779133608152816852651669, 3.21428846901977934002638162414, 3.44592046486286322546217075548, 3.54080698225297808582796089022, 4.05146868372649466566531579941, 5.00632052792353410505653707057, 5.01387005247890360020623258373, 5.38409653934961654966137334031, 5.56827772788063819312214295772, 6.00932208788318772113606105425, 6.26266052959310795509096411371, 6.67935056157105064923058187076, 6.73292998674861013119023952939, 7.22266349263049605087810173676, 7.72461929978765157566273487860
