| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 6·11-s − 15-s − 2·21-s + 25-s − 27-s − 6·33-s + 2·35-s + 8·43-s + 45-s − 2·49-s − 12·53-s + 6·55-s + 18·59-s + 4·61-s + 2·63-s − 4·67-s + 12·71-s − 75-s + 12·77-s + 81-s + 6·99-s − 10·103-s − 2·105-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.80·11-s − 0.258·15-s − 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.04·33-s + 0.338·35-s + 1.21·43-s + 0.149·45-s − 2/7·49-s − 1.64·53-s + 0.809·55-s + 2.34·59-s + 0.512·61-s + 0.251·63-s − 0.488·67-s + 1.42·71-s − 0.115·75-s + 1.36·77-s + 1/9·81-s + 0.603·99-s − 0.985·103-s − 0.195·105-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.963907807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.963907807\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.141241154029135032259294624053, −8.609298928202332924153573545631, −8.111123297949343578743720333764, −7.62278521410384067925608890140, −6.93159405149276696220808232612, −6.63621551256985941580492577404, −6.20367239541730683512107851461, −5.56354189044085788017222900632, −5.17790188564623764239537996870, −4.47799600643106816674367720352, −4.05726476535751679542914610079, −3.46610539182281159391386066816, −2.46858524619923688484799519620, −1.68298604675044022991253635007, −1.01603419970345686879799970894,
1.01603419970345686879799970894, 1.68298604675044022991253635007, 2.46858524619923688484799519620, 3.46610539182281159391386066816, 4.05726476535751679542914610079, 4.47799600643106816674367720352, 5.17790188564623764239537996870, 5.56354189044085788017222900632, 6.20367239541730683512107851461, 6.63621551256985941580492577404, 6.93159405149276696220808232612, 7.62278521410384067925608890140, 8.111123297949343578743720333764, 8.609298928202332924153573545631, 9.141241154029135032259294624053
