| L(s) = 1 | + 1.59e4·5-s − 9.82e4·7-s + 1.63e6·11-s + 2.23e7·13-s + 1.22e8·17-s + 7.42e7·19-s − 1.06e9·23-s − 9.66e8·25-s + 5.60e9·29-s − 2.16e9·31-s − 1.56e9·35-s − 5.95e9·37-s − 2.16e10·41-s + 6.11e10·43-s + 1.36e11·47-s − 8.72e10·49-s − 5.57e8·53-s + 2.59e10·55-s − 3.02e11·59-s − 1.90e11·61-s + 3.55e11·65-s + 9.18e11·67-s − 1.08e12·71-s − 7.72e10·73-s − 1.60e11·77-s − 2.62e12·79-s + 3.26e12·83-s + ⋯ |
| L(s) = 1 | + 0.456·5-s − 0.315·7-s + 0.277·11-s + 1.28·13-s + 1.23·17-s + 0.362·19-s − 1.50·23-s − 0.791·25-s + 1.75·29-s − 0.437·31-s − 0.143·35-s − 0.381·37-s − 0.712·41-s + 1.47·43-s + 1.84·47-s − 0.900·49-s − 0.00345·53-s + 0.126·55-s − 0.932·59-s − 0.473·61-s + 0.584·65-s + 1.24·67-s − 1.00·71-s − 0.0597·73-s − 0.0875·77-s − 1.21·79-s + 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.748892706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.748892706\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 15936 T + p^{13} T^{2} \) |
| 7 | \( 1 + 14036 p T + p^{13} T^{2} \) |
| 11 | \( 1 - 148224 p T + p^{13} T^{2} \) |
| 13 | \( 1 - 22316306 T + p^{13} T^{2} \) |
| 17 | \( 1 - 122937984 T + p^{13} T^{2} \) |
| 19 | \( 1 - 74272984 T + p^{13} T^{2} \) |
| 23 | \( 1 + 1069509120 T + p^{13} T^{2} \) |
| 29 | \( 1 - 5607090624 T + p^{13} T^{2} \) |
| 31 | \( 1 + 2162031116 T + p^{13} T^{2} \) |
| 37 | \( 1 + 5959452922 T + p^{13} T^{2} \) |
| 41 | \( 1 + 21676851840 T + p^{13} T^{2} \) |
| 43 | \( 1 - 61101030232 T + p^{13} T^{2} \) |
| 47 | \( 1 - 136471948800 T + p^{13} T^{2} \) |
| 53 | \( 1 + 557015616 T + p^{13} T^{2} \) |
| 59 | \( 1 + 302211949056 T + p^{13} T^{2} \) |
| 61 | \( 1 + 190535454658 T + p^{13} T^{2} \) |
| 67 | \( 1 - 918343123024 T + p^{13} T^{2} \) |
| 71 | \( 1 + 1086593292288 T + p^{13} T^{2} \) |
| 73 | \( 1 + 77275903210 T + p^{13} T^{2} \) |
| 79 | \( 1 + 2624363498636 T + p^{13} T^{2} \) |
| 83 | \( 1 - 3269608182528 T + p^{13} T^{2} \) |
| 89 | \( 1 + 5922230600448 T + p^{13} T^{2} \) |
| 97 | \( 1 - 5340133325582 T + p^{13} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.51072722596483617867598278694, −9.756222393363060880622232255511, −8.665826651135724573801980948049, −7.64239779858999579764009629105, −6.28836967918330449726214979753, −5.65307903097327160650014157279, −4.13149007919237615320932063673, −3.14127811042593929374103262109, −1.76857505357847837792490069531, −0.76100867238915668260258468391,
0.76100867238915668260258468391, 1.76857505357847837792490069531, 3.14127811042593929374103262109, 4.13149007919237615320932063673, 5.65307903097327160650014157279, 6.28836967918330449726214979753, 7.64239779858999579764009629105, 8.665826651135724573801980948049, 9.756222393363060880622232255511, 10.51072722596483617867598278694
