L(s) = 1 | − 2-s + 3-s + 4-s − 4.22·5-s − 6-s + 3.89·7-s − 8-s + 9-s + 4.22·10-s + 1.39·11-s + 12-s + 6.39·13-s − 3.89·14-s − 4.22·15-s + 16-s + 6.62·17-s − 18-s + 19-s − 4.22·20-s + 3.89·21-s − 1.39·22-s + 4.91·23-s − 24-s + 12.8·25-s − 6.39·26-s + 27-s + 3.89·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.88·5-s − 0.408·6-s + 1.47·7-s − 0.353·8-s + 0.333·9-s + 1.33·10-s + 0.421·11-s + 0.288·12-s + 1.77·13-s − 1.04·14-s − 1.09·15-s + 0.250·16-s + 1.60·17-s − 0.235·18-s + 0.229·19-s − 0.944·20-s + 0.849·21-s − 0.297·22-s + 1.02·23-s − 0.204·24-s + 2.56·25-s − 1.25·26-s + 0.192·27-s + 0.735·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.070058688\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.070058688\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
good | 5 | \( 1 + 4.22T + 5T^{2} \) |
| 7 | \( 1 - 3.89T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 13 | \( 1 - 6.39T + 13T^{2} \) |
| 17 | \( 1 - 6.62T + 17T^{2} \) |
| 23 | \( 1 - 4.91T + 23T^{2} \) |
| 29 | \( 1 - 5.17T + 29T^{2} \) |
| 31 | \( 1 - 9.08T + 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 + 8.33T + 41T^{2} \) |
| 43 | \( 1 + 2.91T + 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 59 | \( 1 - 1.28T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 - 1.72T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 8.59T + 73T^{2} \) |
| 79 | \( 1 - 0.401T + 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 + 3.45T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.230586218665400344964884061636, −7.69076037573186599243789436989, −7.03223782329680238825030599441, −6.15677895149590424130143869140, −4.94807649300013009532307761291, −4.36729743896145632693458892836, −3.45457389963519497606399714419, −3.03020171497537094475545166276, −1.35238237064917503633584467665, −1.01009749164106530204310038226,
1.01009749164106530204310038226, 1.35238237064917503633584467665, 3.03020171497537094475545166276, 3.45457389963519497606399714419, 4.36729743896145632693458892836, 4.94807649300013009532307761291, 6.15677895149590424130143869140, 7.03223782329680238825030599441, 7.69076037573186599243789436989, 8.230586218665400344964884061636