L(s) = 1 | + 2-s + 3-s + 4-s − 2.44·5-s + 6-s − 1.82·7-s + 8-s + 9-s − 2.44·10-s + 0.163·11-s + 12-s − 13-s − 1.82·14-s − 2.44·15-s + 16-s + 6.53·17-s + 18-s + 3.88·19-s − 2.44·20-s − 1.82·21-s + 0.163·22-s − 0.103·23-s + 24-s + 0.996·25-s − 26-s + 27-s − 1.82·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.09·5-s + 0.408·6-s − 0.690·7-s + 0.353·8-s + 0.333·9-s − 0.774·10-s + 0.0493·11-s + 0.288·12-s − 0.277·13-s − 0.488·14-s − 0.632·15-s + 0.250·16-s + 1.58·17-s + 0.235·18-s + 0.890·19-s − 0.547·20-s − 0.398·21-s + 0.0349·22-s − 0.0216·23-s + 0.204·24-s + 0.199·25-s − 0.196·26-s + 0.192·27-s − 0.345·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.009792832\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.009792832\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 + 1.82T + 7T^{2} \) |
| 11 | \( 1 - 0.163T + 11T^{2} \) |
| 17 | \( 1 - 6.53T + 17T^{2} \) |
| 19 | \( 1 - 3.88T + 19T^{2} \) |
| 23 | \( 1 + 0.103T + 23T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 1.24T + 41T^{2} \) |
| 43 | \( 1 + 0.424T + 43T^{2} \) |
| 47 | \( 1 - 7.02T + 47T^{2} \) |
| 53 | \( 1 + 5.21T + 53T^{2} \) |
| 59 | \( 1 + 7.00T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 16.1T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 + 5.61T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 9.39T + 97T^{2} \) |
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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
−7.81996891515984854618882985004, −7.17405342009112298714211703013, −6.57895028380345661294662765902, −5.55841663919581815209803718104, −5.05682941862139229008493876520, −3.99046710032311957131023519783, −3.51361615879586330800619992481, −3.08229127229354163668774046874, −1.98251978921842307358774370432, −0.74269354665933928400946158372,
0.74269354665933928400946158372, 1.98251978921842307358774370432, 3.08229127229354163668774046874, 3.51361615879586330800619992481, 3.99046710032311957131023519783, 5.05682941862139229008493876520, 5.55841663919581815209803718104, 6.57895028380345661294662765902, 7.17405342009112298714211703013, 7.81996891515984854618882985004