| L(s) = 1 | + 6·3-s + 2·5-s + 21·9-s − 11-s + 12·15-s − 4·23-s + 3·25-s + 54·27-s + 10·31-s − 6·33-s − 2·37-s + 42·45-s − 4·47-s − 13·49-s − 26·53-s − 2·55-s + 4·59-s + 32·67-s − 24·69-s + 30·71-s + 18·75-s + 108·81-s + 18·89-s + 60·93-s − 32·97-s − 21·99-s − 32·103-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 0.894·5-s + 7·9-s − 0.301·11-s + 3.09·15-s − 0.834·23-s + 3/5·25-s + 10.3·27-s + 1.79·31-s − 1.04·33-s − 0.328·37-s + 6.26·45-s − 0.583·47-s − 1.85·49-s − 3.57·53-s − 0.269·55-s + 0.520·59-s + 3.90·67-s − 2.88·69-s + 3.56·71-s + 2.07·75-s + 12·81-s + 1.90·89-s + 6.22·93-s − 3.24·97-s − 2.11·99-s − 3.15·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2129600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2129600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.23970432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.23970432\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−7.997108060100772027894387055712, −7.79929638968846771610897877135, −6.73950043879569860448501959504, −6.67194470482055909928367851751, −6.49668472503709804070474649098, −5.26099188052484833696420628416, −5.18236780553163224027614577017, −4.30915459197604469230806454554, −4.07821246743313595622649858247, −3.45273821080139062689150131200, −3.02944359256572950711565258745, −2.77079934605572561635474291231, −2.02645929507657162236826308088, −1.97350758509779452319901265349, −1.18939568880827510373065837222,
1.18939568880827510373065837222, 1.97350758509779452319901265349, 2.02645929507657162236826308088, 2.77079934605572561635474291231, 3.02944359256572950711565258745, 3.45273821080139062689150131200, 4.07821246743313595622649858247, 4.30915459197604469230806454554, 5.18236780553163224027614577017, 5.26099188052484833696420628416, 6.49668472503709804070474649098, 6.67194470482055909928367851751, 6.73950043879569860448501959504, 7.79929638968846771610897877135, 7.997108060100772027894387055712
