| L(s) = 1 | + 4-s + 4·5-s − 6·9-s + 4·11-s + 16-s + 4·20-s + 16·23-s + 2·25-s − 16·31-s − 6·36-s + 12·37-s + 4·44-s − 24·45-s − 16·47-s + 49-s + 12·53-s + 16·55-s + 16·59-s + 64-s + 8·67-s − 16·71-s + 4·80-s + 27·81-s + 36·89-s + 16·92-s + 4·97-s − 24·99-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.78·5-s − 2·9-s + 1.20·11-s + 1/4·16-s + 0.894·20-s + 3.33·23-s + 2/5·25-s − 2.87·31-s − 36-s + 1.97·37-s + 0.603·44-s − 3.57·45-s − 2.33·47-s + 1/7·49-s + 1.64·53-s + 2.15·55-s + 2.08·59-s + 1/8·64-s + 0.977·67-s − 1.89·71-s + 0.447·80-s + 3·81-s + 3.81·89-s + 1.66·92-s + 0.406·97-s − 2.41·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008004 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008004 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.981576126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.981576126\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.18951670915097387570457531436, −7.08827055361374002488782088399, −6.32358215795933556597678864095, −6.19217622188912966190136871625, −5.91665951755465501251112502300, −5.29943015736314161359833373222, −5.21535057877927416914375242112, −4.73057752361113131284783498762, −3.67764363944170895713490289621, −3.51840718185624303027243325096, −2.96567033999897366329822632527, −2.27950576664097092555756241695, −2.18713180950649411226712364496, −1.38757950676875077475246877875, −0.72883362867603820321005479251,
0.72883362867603820321005479251, 1.38757950676875077475246877875, 2.18713180950649411226712364496, 2.27950576664097092555756241695, 2.96567033999897366329822632527, 3.51840718185624303027243325096, 3.67764363944170895713490289621, 4.73057752361113131284783498762, 5.21535057877927416914375242112, 5.29943015736314161359833373222, 5.91665951755465501251112502300, 6.19217622188912966190136871625, 6.32358215795933556597678864095, 7.08827055361374002488782088399, 7.18951670915097387570457531436
