| L(s) = 1 | − 2·5-s + 4·7-s − 4·11-s − 2·13-s − 4·19-s − 4·23-s − 25-s − 12·29-s + 4·31-s − 8·35-s + 4·37-s − 8·41-s − 2·43-s − 6·47-s − 2·49-s + 8·55-s − 18·59-s − 6·61-s + 4·65-s + 2·71-s + 8·73-s − 16·77-s + 10·83-s + 10·89-s − 8·91-s + 8·95-s − 8·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 1.20·11-s − 0.554·13-s − 0.917·19-s − 0.834·23-s − 1/5·25-s − 2.22·29-s + 0.718·31-s − 1.35·35-s + 0.657·37-s − 1.24·41-s − 0.304·43-s − 0.875·47-s − 2/7·49-s + 1.07·55-s − 2.34·59-s − 0.768·61-s + 0.496·65-s + 0.237·71-s + 0.936·73-s − 1.82·77-s + 1.09·83-s + 1.05·89-s − 0.838·91-s + 0.820·95-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7884864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7884864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 74 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 97 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 193 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 185 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 149 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T - 6 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.326723463047085687873598484706, −8.128320124589235164496135080408, −7.88493509405814614931138703535, −7.60549866946732995663456883615, −7.42595359528748473726913601447, −6.71583947814129803912422337665, −6.25156550791743699989971276316, −6.02524000210886314731030402771, −5.19448216081146952410240417784, −5.17041679424179096362347473235, −4.67104635934303571976224188948, −4.43781549507087555676813887884, −3.64519976959489840375774554568, −3.64080229725246043317507492248, −2.80390475250919230060456668213, −2.27902316118139926880313003510, −1.84632802346597977531480041023, −1.37601901331094319663909839859, 0, 0,
1.37601901331094319663909839859, 1.84632802346597977531480041023, 2.27902316118139926880313003510, 2.80390475250919230060456668213, 3.64080229725246043317507492248, 3.64519976959489840375774554568, 4.43781549507087555676813887884, 4.67104635934303571976224188948, 5.17041679424179096362347473235, 5.19448216081146952410240417784, 6.02524000210886314731030402771, 6.25156550791743699989971276316, 6.71583947814129803912422337665, 7.42595359528748473726913601447, 7.60549866946732995663456883615, 7.88493509405814614931138703535, 8.128320124589235164496135080408, 8.326723463047085687873598484706
