| L(s) = 1 | + 2.37·3-s − 5-s + 4.57·7-s + 2.66·9-s − 5.84·11-s − 2.86·13-s − 2.37·15-s − 6.90·17-s + 2.24·19-s + 10.8·21-s − 0.981·23-s + 25-s − 0.798·27-s − 29-s + 4.18·31-s − 13.9·33-s − 4.57·35-s − 2.31·37-s − 6.81·39-s − 7.15·41-s + 4.90·43-s − 2.66·45-s + 6.85·47-s + 13.9·49-s − 16.4·51-s + 6.06·53-s + 5.84·55-s + ⋯ |
| L(s) = 1 | + 1.37·3-s − 0.447·5-s + 1.73·7-s + 0.888·9-s − 1.76·11-s − 0.794·13-s − 0.614·15-s − 1.67·17-s + 0.514·19-s + 2.37·21-s − 0.204·23-s + 0.200·25-s − 0.153·27-s − 0.185·29-s + 0.751·31-s − 2.42·33-s − 0.773·35-s − 0.380·37-s − 1.09·39-s − 1.11·41-s + 0.748·43-s − 0.397·45-s + 0.999·47-s + 1.99·49-s − 2.30·51-s + 0.833·53-s + 0.788·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.37T + 3T^{2} \) |
| 7 | \( 1 - 4.57T + 7T^{2} \) |
| 11 | \( 1 + 5.84T + 11T^{2} \) |
| 13 | \( 1 + 2.86T + 13T^{2} \) |
| 17 | \( 1 + 6.90T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 + 0.981T + 23T^{2} \) |
| 31 | \( 1 - 4.18T + 31T^{2} \) |
| 37 | \( 1 + 2.31T + 37T^{2} \) |
| 41 | \( 1 + 7.15T + 41T^{2} \) |
| 43 | \( 1 - 4.90T + 43T^{2} \) |
| 47 | \( 1 - 6.85T + 47T^{2} \) |
| 53 | \( 1 - 6.06T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 7.30T + 67T^{2} \) |
| 71 | \( 1 - 9.51T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 2.44T + 79T^{2} \) |
| 83 | \( 1 - 9.93T + 83T^{2} \) |
| 89 | \( 1 + 9.41T + 89T^{2} \) |
| 97 | \( 1 + 4.95T + 97T^{2} \) |
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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
−7.40661029086627872859045963264, −7.30769441414601198903303529190, −5.89700338397496606942138349686, −4.92782223823004167695463053586, −4.69664854655379367579202574340, −3.83965515096744355632644825977, −2.70313592440577253321841678252, −2.43588212506185525160994579899, −1.57418109732938428003146248495, 0,
1.57418109732938428003146248495, 2.43588212506185525160994579899, 2.70313592440577253321841678252, 3.83965515096744355632644825977, 4.69664854655379367579202574340, 4.92782223823004167695463053586, 5.89700338397496606942138349686, 7.30769441414601198903303529190, 7.40661029086627872859045963264
