| L(s) = 1 | − 6·3-s − 6·5-s + 15·9-s + 2·11-s + 36·15-s − 18·17-s + 19·25-s − 18·27-s + 28·29-s − 12·33-s − 90·45-s + 42·47-s + 108·51-s − 12·55-s − 40·71-s − 114·75-s + 6·79-s + 9·81-s + 108·85-s − 168·87-s + 30·99-s − 6·103-s − 6·109-s + 23·121-s − 42·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 2.68·5-s + 5·9-s + 0.603·11-s + 9.29·15-s − 4.36·17-s + 19/5·25-s − 3.46·27-s + 5.19·29-s − 2.08·33-s − 13.4·45-s + 6.12·47-s + 15.1·51-s − 1.61·55-s − 4.74·71-s − 13.1·75-s + 0.675·79-s + 81-s + 11.7·85-s − 18.0·87-s + 3.01·99-s − 0.591·103-s − 0.574·109-s + 2.09·121-s − 3.75·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1180535722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1180535722\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 40 T^{2} + 1071 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 - 12 T^{2} - 817 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 + 20 T^{2} - 969 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 21 T + 194 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 44 T^{2} - 873 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 68 T^{2} + 1143 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 + 78 T^{2} + 2363 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 16 T^{2} - 4233 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.94766071715292051610444062677, −6.87607169478758594981400771039, −6.60684072079493615083742566438, −6.59630213191095520268022567076, −6.32928753986726113292741658156, −5.98351030213213962741704584580, −5.75809394673006333103781119239, −5.72429951133565025457385558397, −5.48684870504763657202160028234, −4.89118372745725648568726880885, −4.70293753865458460111147375758, −4.55811037330846508973892701312, −4.46731755205404478596602934595, −4.41642959136690270109458076433, −4.08435937086795423614483069811, −3.86316634223836448301622129572, −3.44318988615782679350205800760, −2.87985696503756992212430191115, −2.79197291810955489598853220523, −2.34005250572410414018158352680, −2.24641975638746342882848886896, −1.12611711738375771055412319527, −1.05195269477210924731832896615, −0.48529871988183481285331901636, −0.27099643900918629041307917982,
0.27099643900918629041307917982, 0.48529871988183481285331901636, 1.05195269477210924731832896615, 1.12611711738375771055412319527, 2.24641975638746342882848886896, 2.34005250572410414018158352680, 2.79197291810955489598853220523, 2.87985696503756992212430191115, 3.44318988615782679350205800760, 3.86316634223836448301622129572, 4.08435937086795423614483069811, 4.41642959136690270109458076433, 4.46731755205404478596602934595, 4.55811037330846508973892701312, 4.70293753865458460111147375758, 4.89118372745725648568726880885, 5.48684870504763657202160028234, 5.72429951133565025457385558397, 5.75809394673006333103781119239, 5.98351030213213962741704584580, 6.32928753986726113292741658156, 6.59630213191095520268022567076, 6.60684072079493615083742566438, 6.87607169478758594981400771039, 6.94766071715292051610444062677
