| L(s) = 1 | + 2-s + 4-s − 2.62·5-s − 7-s + 8-s − 2.62·10-s − 3.52·11-s + 3.76·13-s − 14-s + 16-s − 4.38·17-s + 2.86·19-s − 2.62·20-s − 3.52·22-s − 23-s + 1.89·25-s + 3.76·26-s − 28-s + 4.89·29-s + 6.38·31-s + 32-s − 4.38·34-s + 2.62·35-s + 2.89·37-s + 2.86·38-s − 2.62·40-s + 10.1·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.17·5-s − 0.377·7-s + 0.353·8-s − 0.830·10-s − 1.06·11-s + 1.04·13-s − 0.267·14-s + 0.250·16-s − 1.06·17-s + 0.657·19-s − 0.587·20-s − 0.751·22-s − 0.208·23-s + 0.379·25-s + 0.737·26-s − 0.188·28-s + 0.909·29-s + 1.14·31-s + 0.176·32-s − 0.752·34-s + 0.443·35-s + 0.476·37-s + 0.464·38-s − 0.415·40-s + 1.58·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.005572274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.005572274\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2.62T + 5T^{2} \) |
| 11 | \( 1 + 3.52T + 11T^{2} \) |
| 13 | \( 1 - 3.76T + 13T^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 19 | \( 1 - 2.86T + 19T^{2} \) |
| 29 | \( 1 - 4.89T + 29T^{2} \) |
| 31 | \( 1 - 6.38T + 31T^{2} \) |
| 37 | \( 1 - 2.89T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 6.35T + 43T^{2} \) |
| 47 | \( 1 - 7.49T + 47T^{2} \) |
| 53 | \( 1 - 1.79T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6.77T + 61T^{2} \) |
| 67 | \( 1 + 7.04T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 7.72T + 73T^{2} \) |
| 79 | \( 1 - 5.52T + 79T^{2} \) |
| 83 | \( 1 - 8.11T + 83T^{2} \) |
| 89 | \( 1 - 4.32T + 89T^{2} \) |
| 97 | \( 1 - 0.238T + 97T^{2} \) |
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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
−8.579613119282335532571178666402, −7.86871994784961681010015299001, −7.32854651671487418096170973231, −6.36285556638970861365532197725, −5.74914663878108011773436201418, −4.61987455623823588262244816254, −4.13920879353469205030819522387, −3.19916014668450560593735387582, −2.45381115096225028698021916980, −0.77083713400128663335718594706,
0.77083713400128663335718594706, 2.45381115096225028698021916980, 3.19916014668450560593735387582, 4.13920879353469205030819522387, 4.61987455623823588262244816254, 5.74914663878108011773436201418, 6.36285556638970861365532197725, 7.32854651671487418096170973231, 7.86871994784961681010015299001, 8.579613119282335532571178666402
