L(s) = 1 | + 2·3-s − 3·4-s + 2·5-s + 3·9-s + 4·11-s − 6·12-s + 4·15-s + 5·16-s − 6·20-s + 16·23-s + 3·25-s + 4·27-s − 16·31-s + 8·33-s − 9·36-s + 12·37-s − 12·44-s + 6·45-s − 16·47-s + 10·48-s − 14·49-s + 12·53-s + 8·55-s − 24·59-s − 12·60-s − 3·64-s − 8·67-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 3/2·4-s + 0.894·5-s + 9-s + 1.20·11-s − 1.73·12-s + 1.03·15-s + 5/4·16-s − 1.34·20-s + 3.33·23-s + 3/5·25-s + 0.769·27-s − 2.87·31-s + 1.39·33-s − 3/2·36-s + 1.97·37-s − 1.80·44-s + 0.894·45-s − 2.33·47-s + 1.44·48-s − 2·49-s + 1.64·53-s + 1.07·55-s − 3.12·59-s − 1.54·60-s − 3/8·64-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4601025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4601025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.576012997\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.576012997\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−7.54085143340292071159442984327, −7.02617658723181735563522650723, −6.48654966834348628333399350819, −6.24012153697644363528550581310, −5.66506514104694592652509606044, −4.94734620594985484281629006959, −4.88823595164803979266926983864, −4.59542807880365420672817957846, −3.75894385007898905483554507552, −3.56799192480000330057521215275, −3.08835967698158237163354511559, −2.63990873433467548408847720531, −1.62272537170773867336279963143, −1.53827946418072491307494025034, −0.66633821317753351700694228489,
0.66633821317753351700694228489, 1.53827946418072491307494025034, 1.62272537170773867336279963143, 2.63990873433467548408847720531, 3.08835967698158237163354511559, 3.56799192480000330057521215275, 3.75894385007898905483554507552, 4.59542807880365420672817957846, 4.88823595164803979266926983864, 4.94734620594985484281629006959, 5.66506514104694592652509606044, 6.24012153697644363528550581310, 6.48654966834348628333399350819, 7.02617658723181735563522650723, 7.54085143340292071159442984327