| L(s) = 1 | + 2.18e3·3-s + 2.29e5·7-s + 3.18e6·9-s − 5.41e7·19-s + 5.01e8·21-s + 1.22e9·25-s + 3.48e9·27-s − 1.53e10·31-s + 2.66e10·37-s − 2.49e10·43-s − 4.43e10·49-s − 1.18e11·57-s + 4.24e11·61-s + 7.31e11·63-s + 1.15e12·67-s − 4.33e12·73-s + 2.66e12·75-s + 3.26e12·79-s + 2.54e12·81-s − 3.35e13·93-s − 3.88e13·103-s − 2.62e13·109-s + 5.83e13·111-s − 3.45e13·121-s + 127-s − 5.45e13·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.736·7-s + 2·9-s − 0.264·19-s + 1.27·21-s + 25-s + 1.73·27-s − 3.10·31-s + 1.70·37-s − 0.602·43-s − 0.457·49-s − 0.457·57-s + 1.05·61-s + 1.47·63-s + 1.55·67-s − 3.35·73-s + 1.73·75-s + 1.51·79-s + 81-s − 5.37·93-s − 3.20·103-s − 1.49·109-s + 2.96·111-s − 121-s − 1.04·129-s − 0.194·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(6.386815435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.386815435\) |
| \(L(\frac{15}{2})\) |
|
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 | $C_2$ | \( 1 - p^{7} T + p^{13} T^{2} \) |
| 7 | $C_2$ | \( 1 - 229315 T + p^{13} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p^{13} T^{2} + p^{26} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{13} T^{2} + p^{26} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 17755915 T + p^{13} T^{2} )( 1 + 17755915 T + p^{13} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p^{13} T^{2} + p^{26} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 177395393 T + p^{13} T^{2} )( 1 + 231547688 T + p^{13} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p^{13} T^{2} + p^{26} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5479470367 T + p^{13} T^{2} )( 1 + 9862529036 T + p^{13} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 27377427169 T + p^{13} T^{2} )( 1 + 705453830 T + p^{13} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12480269243 T + p^{13} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{13} T^{2} + p^{26} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{13} T^{2} + p^{26} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{13} T^{2} + p^{26} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 595391378401 T + p^{13} T^{2} )( 1 + 171215488093 T + p^{13} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 1380765177776 T + p^{13} T^{2} )( 1 + 226871917261 T + p^{13} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1848505355417 T + p^{13} T^{2} )( 1 + 2490400654570 T + p^{13} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4084540747547 T + p^{13} T^{2} )( 1 + 820761767284 T + p^{13} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{13} T^{2} + p^{26} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12352800515314 T + p^{13} T^{2} )( 1 + 12352800515314 T + p^{13} T^{2} ) \) |
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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
−12.02801818221388409967704324832, −11.08501333294885044729968864134, −11.02075968607003169966851003349, −10.11439326840528477472197127064, −9.622494222063266255706459008508, −8.923167488288520932066169035714, −8.834926159410580721636323490532, −7.936347349030348561613666604243, −7.77681531533028136250750851832, −7.06977740417249078090601446419, −6.51994949522374417205767819444, −5.46940821354357475728118928432, −4.99620027912677890653264314878, −3.98252220332784173983308771273, −3.92160095006728802054511888271, −2.91421459357251496891128434419, −2.55950062884443633123630166246, −1.68206492021989822372669263774, −1.48896485406724744026563786396, −0.47237148835267380232806677165,
0.47237148835267380232806677165, 1.48896485406724744026563786396, 1.68206492021989822372669263774, 2.55950062884443633123630166246, 2.91421459357251496891128434419, 3.92160095006728802054511888271, 3.98252220332784173983308771273, 4.99620027912677890653264314878, 5.46940821354357475728118928432, 6.51994949522374417205767819444, 7.06977740417249078090601446419, 7.77681531533028136250750851832, 7.936347349030348561613666604243, 8.834926159410580721636323490532, 8.923167488288520932066169035714, 9.622494222063266255706459008508, 10.11439326840528477472197127064, 11.02075968607003169966851003349, 11.08501333294885044729968864134, 12.02801818221388409967704324832
