| L(s) = 1 | − 2-s − 2.41·3-s − 4-s + 5-s + 2.41·6-s − 2·7-s + 3·8-s + 2.82·9-s − 10-s + 11-s + 2.41·12-s − 0.414·13-s + 2·14-s − 2.41·15-s − 16-s − 4.82·17-s − 2.82·18-s + 8.65·19-s − 20-s + 4.82·21-s − 22-s + 6.07·23-s − 7.24·24-s + 25-s + 0.414·26-s + 0.414·27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.39·3-s − 0.5·4-s + 0.447·5-s + 0.985·6-s − 0.755·7-s + 1.06·8-s + 0.942·9-s − 0.316·10-s + 0.301·11-s + 0.696·12-s − 0.114·13-s + 0.534·14-s − 0.623·15-s − 0.250·16-s − 1.17·17-s − 0.666·18-s + 1.98·19-s − 0.223·20-s + 1.05·21-s − 0.213·22-s + 1.26·23-s − 1.47·24-s + 0.200·25-s + 0.0812·26-s + 0.0797·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 2 | \( 1 + T + 2T^{2} \) |
| 3 | \( 1 + 2.41T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 0.414T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 8.65T + 19T^{2} \) |
| 23 | \( 1 - 6.07T + 23T^{2} \) |
| 29 | \( 1 + 2.17T + 29T^{2} \) |
| 31 | \( 1 - 0.171T + 31T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 1.17T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 + 4.75T + 67T^{2} \) |
| 71 | \( 1 + 1.65T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 4.82T + 79T^{2} \) |
| 83 | \( 1 + 2.75T + 83T^{2} \) |
| 89 | \( 1 - 4.82T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−9.774868213170081917276976705775, −9.405339853024141132834126292803, −8.365835630055788679543814735904, −7.05629044359739349273215587477, −6.54252808336559886268672965448, −5.32007766231248646305615638013, −4.84513083220428973500859540515, −3.34038387546127322723250631734, −1.34919031080963580864429907392, 0,
1.34919031080963580864429907392, 3.34038387546127322723250631734, 4.84513083220428973500859540515, 5.32007766231248646305615638013, 6.54252808336559886268672965448, 7.05629044359739349273215587477, 8.365835630055788679543814735904, 9.405339853024141132834126292803, 9.774868213170081917276976705775
