| L(s) = 1 | − 2-s + 2·3-s − 2·4-s − 2·6-s − 2·7-s + 3·8-s + 3·9-s − 4·12-s − 6·13-s + 2·14-s + 16-s + 2·17-s − 3·18-s + 4·19-s − 4·21-s − 4·23-s + 6·24-s + 6·26-s + 4·27-s + 4·28-s + 2·29-s − 2·32-s − 2·34-s − 6·36-s − 6·37-s − 4·38-s − 12·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 4-s − 0.816·6-s − 0.755·7-s + 1.06·8-s + 9-s − 1.15·12-s − 1.66·13-s + 0.534·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s + 0.917·19-s − 0.872·21-s − 0.834·23-s + 1.22·24-s + 1.17·26-s + 0.769·27-s + 0.755·28-s + 0.371·29-s − 0.353·32-s − 0.342·34-s − 36-s − 0.986·37-s − 0.648·38-s − 1.92·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 53 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 111 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 18 T + 210 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 162 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 149 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 14 T + 222 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 30 T + 414 T^{2} + 30 p T^{3} + p^{2} T^{4} \) |
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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
−7.72584252936632828977318664831, −7.49152562785980453548580382487, −6.93764600361196215354123343632, −6.77766428427882418282722356789, −6.28619686424788528994782746280, −5.82233868918725640029880988959, −5.29440222628224669673302493868, −5.25311088307886077093229285075, −4.65609301551566182806146511465, −4.26608370404525529981037676779, −4.13592910208986982968734008314, −3.55039110071424185028343038232, −3.29243648621660955026242074366, −2.65797005307483326133354947035, −2.57042164497921870956891028208, −2.08599826982338156044202912882, −1.17017691501324022322525447781, −1.14598059022713544528546909865, 0, 0,
1.14598059022713544528546909865, 1.17017691501324022322525447781, 2.08599826982338156044202912882, 2.57042164497921870956891028208, 2.65797005307483326133354947035, 3.29243648621660955026242074366, 3.55039110071424185028343038232, 4.13592910208986982968734008314, 4.26608370404525529981037676779, 4.65609301551566182806146511465, 5.25311088307886077093229285075, 5.29440222628224669673302493868, 5.82233868918725640029880988959, 6.28619686424788528994782746280, 6.77766428427882418282722356789, 6.93764600361196215354123343632, 7.49152562785980453548580382487, 7.72584252936632828977318664831
