| L(s) = 1 | + 3-s + 4.21·5-s − 5.13·7-s + 9-s − 1.59·11-s + 5.05·13-s + 4.21·15-s + 0.754·17-s − 1.66·19-s − 5.13·21-s + 6.04·23-s + 12.7·25-s + 27-s − 4.24·29-s − 1.07·31-s − 1.59·33-s − 21.6·35-s + 4.94·37-s + 5.05·39-s + 3.04·41-s + 12.5·43-s + 4.21·45-s + 9.35·47-s + 19.3·49-s + 0.754·51-s − 8.52·53-s − 6.70·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.88·5-s − 1.94·7-s + 0.333·9-s − 0.479·11-s + 1.40·13-s + 1.08·15-s + 0.183·17-s − 0.382·19-s − 1.12·21-s + 1.25·23-s + 2.54·25-s + 0.192·27-s − 0.788·29-s − 0.192·31-s − 0.277·33-s − 3.65·35-s + 0.813·37-s + 0.809·39-s + 0.475·41-s + 1.91·43-s + 0.627·45-s + 1.36·47-s + 2.77·49-s + 0.105·51-s − 1.17·53-s − 0.903·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.669339495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.669339495\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 4.21T + 5T^{2} \) |
| 7 | \( 1 + 5.13T + 7T^{2} \) |
| 11 | \( 1 + 1.59T + 11T^{2} \) |
| 13 | \( 1 - 5.05T + 13T^{2} \) |
| 17 | \( 1 - 0.754T + 17T^{2} \) |
| 19 | \( 1 + 1.66T + 19T^{2} \) |
| 23 | \( 1 - 6.04T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 1.07T + 31T^{2} \) |
| 37 | \( 1 - 4.94T + 37T^{2} \) |
| 41 | \( 1 - 3.04T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 9.35T + 47T^{2} \) |
| 53 | \( 1 + 8.52T + 53T^{2} \) |
| 59 | \( 1 + 3.98T + 59T^{2} \) |
| 61 | \( 1 - 8.18T + 61T^{2} \) |
| 67 | \( 1 + 4.99T + 67T^{2} \) |
| 71 | \( 1 - 6.73T + 71T^{2} \) |
| 73 | \( 1 + 6.06T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 5.95T + 83T^{2} \) |
| 89 | \( 1 - 7.83T + 89T^{2} \) |
| 97 | \( 1 - 3.42T + 97T^{2} \) |
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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
−9.137053761083641047094118599598, −8.875510139957816296798559600901, −7.46488823691065041626710244673, −6.58274312766228868252459762560, −6.04108526370861104996526200426, −5.49108845893276005010803699238, −4.02814893399526549694690099100, −3.01219827919237535795856060293, −2.46217387955128702358148978148, −1.12419056454476401417286890502,
1.12419056454476401417286890502, 2.46217387955128702358148978148, 3.01219827919237535795856060293, 4.02814893399526549694690099100, 5.49108845893276005010803699238, 6.04108526370861104996526200426, 6.58274312766228868252459762560, 7.46488823691065041626710244673, 8.875510139957816296798559600901, 9.137053761083641047094118599598
