L(s) = 1 | + 2-s − 2·3-s − 4-s − 2·6-s − 4·7-s − 3·8-s + 9-s + 4·11-s + 2·12-s − 4·14-s − 16-s + 8·17-s + 18-s + 4·19-s + 8·21-s + 4·22-s − 2·23-s + 6·24-s + 4·27-s + 4·28-s − 6·29-s + 31-s + 5·32-s − 8·33-s + 8·34-s − 36-s + 4·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.816·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s + 1.20·11-s + 0.577·12-s − 1.06·14-s − 1/4·16-s + 1.94·17-s + 0.235·18-s + 0.917·19-s + 1.74·21-s + 0.852·22-s − 0.417·23-s + 1.22·24-s + 0.769·27-s + 0.755·28-s − 1.11·29-s + 0.179·31-s + 0.883·32-s − 1.39·33-s + 1.37·34-s − 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9454718718\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9454718718\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−10.12480892911710644544204331348, −9.692472166227338706592489359972, −8.846992762987307782653877670092, −7.43115311805805500208954053517, −6.33626940076817063547432566790, −5.88299041894911118235279795178, −5.11890327008721802890234062200, −3.81927466922608009941657653944, −3.21813341037386174475685321406, −0.76673337758401995531165024074,
0.76673337758401995531165024074, 3.21813341037386174475685321406, 3.81927466922608009941657653944, 5.11890327008721802890234062200, 5.88299041894911118235279795178, 6.33626940076817063547432566790, 7.43115311805805500208954053517, 8.846992762987307782653877670092, 9.692472166227338706592489359972, 10.12480892911710644544204331348