L(s) = 1 | + (−1.53 + 0.5i)3-s + (1.30 − 0.951i)9-s + (−0.951 + 0.309i)11-s + (−0.5 − 0.363i)17-s + (0.587 − 0.190i)19-s + (−0.309 − 0.951i)25-s + (−0.587 + 0.809i)27-s + (1.30 − 0.951i)33-s + (0.5 + 1.53i)41-s − 1.61i·43-s + (−0.809 − 0.587i)49-s + (0.951 + 0.309i)51-s + (−0.809 + 0.587i)57-s + (0.587 + 0.190i)59-s − 0.618i·67-s + ⋯ |
L(s) = 1 | + (−1.53 + 0.5i)3-s + (1.30 − 0.951i)9-s + (−0.951 + 0.309i)11-s + (−0.5 − 0.363i)17-s + (0.587 − 0.190i)19-s + (−0.309 − 0.951i)25-s + (−0.587 + 0.809i)27-s + (1.30 − 0.951i)33-s + (0.5 + 1.53i)41-s − 1.61i·43-s + (−0.809 − 0.587i)49-s + (0.951 + 0.309i)51-s + (−0.809 + 0.587i)57-s + (0.587 + 0.190i)59-s − 0.618i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5267352237\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5267352237\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (0.951 - 0.309i)T \) |
good | 3 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + 1.61iT - T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + 0.618iT - T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 - 1.61T + T^{2} \) |
| 97 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
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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
−9.076676486904708761161492402474, −8.037570223526660015805431691984, −7.25101873106452368431996014946, −6.42568485269402475681590059808, −5.79846088411510259110809904688, −4.91926786498934091688760943018, −4.60485372853726525882644099636, −3.39554511188462129942097402418, −2.14504761880064535210145622157, −0.50483085446331176208181337249,
1.00661743386962864456399335901, 2.22693331712534398912319082525, 3.51220835922272268717551590742, 4.66809837939477596098331693124, 5.37056276458265135724687768518, 5.91506250182217909759973215063, 6.67279777179898954220948466023, 7.44473854578433523202079964834, 8.052046731600697002194008237818, 9.111162815866615382103920401759