| L(s) = 1 | − 3·3-s + 2·7-s + 6·9-s − 10·13-s − 4·19-s − 6·21-s + 25-s − 9·27-s + 8·31-s − 12·37-s + 30·39-s − 4·43-s + 3·49-s + 12·57-s + 8·61-s + 12·63-s + 16·67-s + 12·73-s − 3·75-s − 6·79-s + 9·81-s − 20·91-s − 24·93-s + 26·97-s + 26·103-s + 10·109-s + 36·111-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2·9-s − 2.77·13-s − 0.917·19-s − 1.30·21-s + 1/5·25-s − 1.73·27-s + 1.43·31-s − 1.97·37-s + 4.80·39-s − 0.609·43-s + 3/7·49-s + 1.58·57-s + 1.02·61-s + 1.51·63-s + 1.95·67-s + 1.40·73-s − 0.346·75-s − 0.675·79-s + 81-s − 2.09·91-s − 2.48·93-s + 2.63·97-s + 2.56·103-s + 0.957·109-s + 3.41·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 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
−8.033082332859894178672793368914, −7.42695240476731979776224139494, −7.13289292005793470052045653622, −6.74907559299019305057473208578, −6.32900819422385960555488032719, −5.74978286836860489420343080410, −5.09619695718514137207067111766, −4.93936350214085093868248089925, −4.73560872492356455619359024880, −4.06954744861732417689986542370, −3.30527145306019394907962235588, −2.25342723502104844991389935790, −2.07628009905382843609630348286, −0.887538960576224120433585893779, 0,
0.887538960576224120433585893779, 2.07628009905382843609630348286, 2.25342723502104844991389935790, 3.30527145306019394907962235588, 4.06954744861732417689986542370, 4.73560872492356455619359024880, 4.93936350214085093868248089925, 5.09619695718514137207067111766, 5.74978286836860489420343080410, 6.32900819422385960555488032719, 6.74907559299019305057473208578, 7.13289292005793470052045653622, 7.42695240476731979776224139494, 8.033082332859894178672793368914
