| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 14-s + 16-s + 9·17-s + 3·23-s − 4·25-s + 28-s − 5·31-s + 32-s + 9·34-s − 6·41-s + 3·46-s + 15·47-s − 11·49-s − 4·50-s + 56-s − 5·62-s + 64-s + 9·68-s + 9·71-s − 2·73-s + 7·79-s − 6·82-s − 18·89-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.267·14-s + 1/4·16-s + 2.18·17-s + 0.625·23-s − 4/5·25-s + 0.188·28-s − 0.898·31-s + 0.176·32-s + 1.54·34-s − 0.937·41-s + 0.442·46-s + 2.18·47-s − 1.57·49-s − 0.565·50-s + 0.133·56-s − 0.635·62-s + 1/8·64-s + 1.09·68-s + 1.06·71-s − 0.234·73-s + 0.787·79-s − 0.662·82-s − 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.601887063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.601887063\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.777596936184074189053978544763, −9.222740086654256839512017000331, −8.470545583140268981389114236076, −8.093100953556833625661535197205, −7.51197242355776102235105975842, −7.19222651552160093580087210526, −6.53288394565264996429432829638, −5.74567078119544244539232522497, −5.54893484507205759807392682711, −4.96563908642611843355416265431, −4.25829695879671379745689291849, −3.55140055252351156032517594032, −3.13768544802424321633852667756, −2.14736828982118610633305975944, −1.22938143626373056438740034087,
1.22938143626373056438740034087, 2.14736828982118610633305975944, 3.13768544802424321633852667756, 3.55140055252351156032517594032, 4.25829695879671379745689291849, 4.96563908642611843355416265431, 5.54893484507205759807392682711, 5.74567078119544244539232522497, 6.53288394565264996429432829638, 7.19222651552160093580087210526, 7.51197242355776102235105975842, 8.093100953556833625661535197205, 8.470545583140268981389114236076, 9.222740086654256839512017000331, 9.777596936184074189053978544763
