| L(s) = 1 | − 3-s + 7-s − 3·11-s − 7·13-s + 3·17-s − 7·19-s − 21-s + 2·23-s − 2·27-s + 6·29-s − 7·31-s + 3·33-s + 8·37-s + 7·39-s − 9·41-s + 4·43-s − 18·47-s − 8·49-s − 3·51-s − 12·53-s + 7·57-s + 18·59-s + 7·61-s + 4·67-s − 2·69-s − 3·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 0.904·11-s − 1.94·13-s + 0.727·17-s − 1.60·19-s − 0.218·21-s + 0.417·23-s − 0.384·27-s + 1.11·29-s − 1.25·31-s + 0.522·33-s + 1.31·37-s + 1.12·39-s − 1.40·41-s + 0.609·43-s − 2.62·47-s − 8/7·49-s − 0.420·51-s − 1.64·53-s + 0.927·57-s + 2.34·59-s + 0.896·61-s + 0.488·67-s − 0.240·69-s − 0.356·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 27 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 97 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 154 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 178 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 7 T + 87 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 97 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T - 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 7 T + 75 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 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
−7.69071184685469354061294145647, −6.95413015573372754058778298065, −6.95170271766643310468278822068, −6.70275900120757947349227622482, −6.07924767341386486278729652293, −5.84898395945617048176919807806, −5.34390539863031697054601689820, −5.21375180861537758944244636791, −4.68841296474468421771286576864, −4.64425381691352674954839442030, −4.22665962882076582644641130028, −3.60251565526635849271839419260, −3.07164535779759277953204527935, −2.97756724107309548524670885971, −2.17650259031641557046713767375, −2.14191274648466875718104320786, −1.57142420745204458670775614313, −0.844184796315044619618450154300, 0, 0,
0.844184796315044619618450154300, 1.57142420745204458670775614313, 2.14191274648466875718104320786, 2.17650259031641557046713767375, 2.97756724107309548524670885971, 3.07164535779759277953204527935, 3.60251565526635849271839419260, 4.22665962882076582644641130028, 4.64425381691352674954839442030, 4.68841296474468421771286576864, 5.21375180861537758944244636791, 5.34390539863031697054601689820, 5.84898395945617048176919807806, 6.07924767341386486278729652293, 6.70275900120757947349227622482, 6.95170271766643310468278822068, 6.95413015573372754058778298065, 7.69071184685469354061294145647
