L(s) = 1 | + 2-s + 0.138·3-s + 4-s + 1.13·5-s + 0.138·6-s − 2.28·7-s + 8-s − 2.98·9-s + 1.13·10-s + 4.66·11-s + 0.138·12-s + 1.82·13-s − 2.28·14-s + 0.157·15-s + 16-s + 5.36·17-s − 2.98·18-s − 1.97·19-s + 1.13·20-s − 0.316·21-s + 4.66·22-s + 4.84·23-s + 0.138·24-s − 3.70·25-s + 1.82·26-s − 0.828·27-s − 2.28·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.0799·3-s + 0.5·4-s + 0.509·5-s + 0.0565·6-s − 0.864·7-s + 0.353·8-s − 0.993·9-s + 0.360·10-s + 1.40·11-s + 0.0399·12-s + 0.505·13-s − 0.611·14-s + 0.0407·15-s + 0.250·16-s + 1.30·17-s − 0.702·18-s − 0.454·19-s + 0.254·20-s − 0.0691·21-s + 0.993·22-s + 1.00·23-s + 0.0282·24-s − 0.740·25-s + 0.357·26-s − 0.159·27-s − 0.432·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.907040517\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.907040517\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 - 0.138T + 3T^{2} \) |
| 5 | \( 1 - 1.13T + 5T^{2} \) |
| 7 | \( 1 + 2.28T + 7T^{2} \) |
| 11 | \( 1 - 4.66T + 11T^{2} \) |
| 13 | \( 1 - 1.82T + 13T^{2} \) |
| 17 | \( 1 - 5.36T + 17T^{2} \) |
| 19 | \( 1 + 1.97T + 19T^{2} \) |
| 23 | \( 1 - 4.84T + 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 - 2.93T + 31T^{2} \) |
| 37 | \( 1 - 4.76T + 37T^{2} \) |
| 41 | \( 1 - 3.64T + 41T^{2} \) |
| 43 | \( 1 - 6.16T + 43T^{2} \) |
| 47 | \( 1 - 0.388T + 47T^{2} \) |
| 53 | \( 1 + 0.525T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 0.842T + 71T^{2} \) |
| 73 | \( 1 + 7.06T + 73T^{2} \) |
| 79 | \( 1 + 3.58T + 79T^{2} \) |
| 83 | \( 1 - 5.10T + 83T^{2} \) |
| 89 | \( 1 + 0.209T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−9.450600022834646021008489965083, −8.825872767492219322956076818522, −7.81382295631030536021962537719, −6.71553526045408570603546910801, −6.11346296824963473896146919489, −5.58174654338967485605190250995, −4.30558276355353341170158095336, −3.41490350991213956209191765577, −2.65934675482563854788130290703, −1.18332114578689423803518998378,
1.18332114578689423803518998378, 2.65934675482563854788130290703, 3.41490350991213956209191765577, 4.30558276355353341170158095336, 5.58174654338967485605190250995, 6.11346296824963473896146919489, 6.71553526045408570603546910801, 7.81382295631030536021962537719, 8.825872767492219322956076818522, 9.450600022834646021008489965083