| L(s) = 1 | + 4·5-s − 2·11-s − 4·17-s − 4·19-s + 8·25-s + 14·29-s + 16·31-s − 10·37-s + 2·43-s + 24·47-s − 49-s − 2·53-s − 8·55-s − 16·59-s + 12·61-s − 6·67-s − 20·79-s − 9·81-s + 20·83-s − 16·85-s − 16·95-s − 4·97-s + 12·101-s − 10·107-s − 6·109-s + 8·113-s + 2·121-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.603·11-s − 0.970·17-s − 0.917·19-s + 8/5·25-s + 2.59·29-s + 2.87·31-s − 1.64·37-s + 0.304·43-s + 3.50·47-s − 1/7·49-s − 0.274·53-s − 1.07·55-s − 2.08·59-s + 1.53·61-s − 0.733·67-s − 2.25·79-s − 81-s + 2.19·83-s − 1.73·85-s − 1.64·95-s − 0.406·97-s + 1.19·101-s − 0.966·107-s − 0.574·109-s + 0.752·113-s + 2/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.317027439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.317027439\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.09175911822384893501823113793, −10.67805717949875776131921745508, −10.35670127096813347327475871805, −10.07397152515153498375012293207, −9.702580643477614558712439334401, −8.965539802145006168766497355833, −8.589862870716613432622908501128, −8.488434702398828304283790782527, −7.63535624886946463675416202902, −7.03215059120741030488718027095, −6.43597228751578873118889251037, −6.28252593000336441674969014597, −5.76115276136681354195826822349, −5.11215983379857543480039693700, −4.58390955543127375685342795580, −4.22433895433586790162957033536, −2.93871229315693150871133454447, −2.60887290518471560943107486435, −2.03452628303811696987370072213, −1.01412491854280694197179319487,
1.01412491854280694197179319487, 2.03452628303811696987370072213, 2.60887290518471560943107486435, 2.93871229315693150871133454447, 4.22433895433586790162957033536, 4.58390955543127375685342795580, 5.11215983379857543480039693700, 5.76115276136681354195826822349, 6.28252593000336441674969014597, 6.43597228751578873118889251037, 7.03215059120741030488718027095, 7.63535624886946463675416202902, 8.488434702398828304283790782527, 8.589862870716613432622908501128, 8.965539802145006168766497355833, 9.702580643477614558712439334401, 10.07397152515153498375012293207, 10.35670127096813347327475871805, 10.67805717949875776131921745508, 11.09175911822384893501823113793
