| L(s) = 1 | + 2·3-s − 2·5-s + 4·7-s + 3·9-s + 2·11-s − 4·15-s − 8·17-s + 8·19-s + 8·21-s + 8·23-s + 3·25-s + 4·27-s − 4·29-s + 4·33-s − 8·35-s + 12·37-s + 4·41-s + 12·43-s − 6·45-s + 8·47-s + 6·49-s − 16·51-s − 4·53-s − 4·55-s + 16·57-s + 8·59-s − 12·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1.51·7-s + 9-s + 0.603·11-s − 1.03·15-s − 1.94·17-s + 1.83·19-s + 1.74·21-s + 1.66·23-s + 3/5·25-s + 0.769·27-s − 0.742·29-s + 0.696·33-s − 1.35·35-s + 1.97·37-s + 0.624·41-s + 1.82·43-s − 0.894·45-s + 1.16·47-s + 6/7·49-s − 2.24·51-s − 0.549·53-s − 0.539·55-s + 2.11·57-s + 1.04·59-s − 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6969600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6969600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.458534352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.458534352\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.202009062815897747967729791089, −8.712467036295104978643198968595, −8.257506283693475089293387976944, −7.83598474978289221756448971831, −7.52655919267472691076321109109, −7.50563407396158848205176707776, −6.81228728071176371334076230445, −6.72106798873757498080537612128, −5.73128040437547016483917302648, −5.70959029180887500582602075568, −4.83487545341611241754704708042, −4.62114061221701144897874031811, −4.26215113462449310436068329124, −4.03053086960840806852089525544, −3.18423974469110956922011270208, −3.06487364219083738510405362355, −2.34941232074031148075831298948, −1.98820282892135820732756833986, −1.15749125850269882240041475256, −0.844507171755473524541580590000,
0.844507171755473524541580590000, 1.15749125850269882240041475256, 1.98820282892135820732756833986, 2.34941232074031148075831298948, 3.06487364219083738510405362355, 3.18423974469110956922011270208, 4.03053086960840806852089525544, 4.26215113462449310436068329124, 4.62114061221701144897874031811, 4.83487545341611241754704708042, 5.70959029180887500582602075568, 5.73128040437547016483917302648, 6.72106798873757498080537612128, 6.81228728071176371334076230445, 7.50563407396158848205176707776, 7.52655919267472691076321109109, 7.83598474978289221756448971831, 8.257506283693475089293387976944, 8.712467036295104978643198968595, 9.202009062815897747967729791089
