| L(s) = 1 | + 2·3-s + 2·5-s + 2·7-s + 3·9-s + 4·13-s + 4·15-s − 2·17-s + 12·19-s + 4·21-s + 2·23-s + 3·25-s + 4·27-s − 10·29-s + 2·31-s + 4·35-s + 2·37-s + 8·39-s + 6·41-s + 4·43-s + 6·45-s + 4·47-s − 3·49-s − 4·51-s + 2·53-s + 24·57-s − 2·59-s + 6·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s + 1.10·13-s + 1.03·15-s − 0.485·17-s + 2.75·19-s + 0.872·21-s + 0.417·23-s + 3/5·25-s + 0.769·27-s − 1.85·29-s + 0.359·31-s + 0.676·35-s + 0.328·37-s + 1.28·39-s + 0.937·41-s + 0.609·43-s + 0.894·45-s + 0.583·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s + 3.17·57-s − 0.260·59-s + 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.764644392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.764644392\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 81 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 73 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 80 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 89 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 111 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 141 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 132 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 197 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 314 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 10 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.200634669438440961362893289719, −8.104975851131140692663251958422, −7.67038220229458437581256168661, −7.32305860495308234001708396771, −6.88612180177375309388738360845, −6.86284105408132507350035227485, −5.98624564788377199296581047810, −5.84447752046096597333364242467, −5.37403304571273071400979093884, −5.27606114813668981785376092263, −4.51658444697396938229746598583, −4.36869993005737002778636514323, −3.66190094936129529957273659331, −3.54957268188513932688014032935, −2.88736651361336046669144505115, −2.74679469783100957452263129259, −1.95723560914404829761241243371, −1.82172341760027644092478262511, −1.10247808635851129708709328847, −0.871214173717780393700615575167,
0.871214173717780393700615575167, 1.10247808635851129708709328847, 1.82172341760027644092478262511, 1.95723560914404829761241243371, 2.74679469783100957452263129259, 2.88736651361336046669144505115, 3.54957268188513932688014032935, 3.66190094936129529957273659331, 4.36869993005737002778636514323, 4.51658444697396938229746598583, 5.27606114813668981785376092263, 5.37403304571273071400979093884, 5.84447752046096597333364242467, 5.98624564788377199296581047810, 6.86284105408132507350035227485, 6.88612180177375309388738360845, 7.32305860495308234001708396771, 7.67038220229458437581256168661, 8.104975851131140692663251958422, 8.200634669438440961362893289719
