| L(s) = 1 | + 1.73·3-s + 3.41·5-s + 0.00658·9-s + 1.67·11-s − 6.13·13-s + 5.91·15-s − 3.78·17-s − 3.86·19-s − 5.49·23-s + 6.65·25-s − 5.19·27-s − 0.172·29-s − 7.15·31-s + 2.90·33-s − 4.80·37-s − 10.6·39-s − 41-s + 0.845·43-s + 0.0224·45-s − 4.65·47-s − 6.57·51-s + 4.46·53-s + 5.72·55-s − 6.70·57-s + 3.38·59-s − 0.643·61-s − 20.9·65-s + ⋯ |
| L(s) = 1 | + 1.00·3-s + 1.52·5-s + 0.00219·9-s + 0.505·11-s − 1.70·13-s + 1.52·15-s − 0.919·17-s − 0.887·19-s − 1.14·23-s + 1.33·25-s − 0.998·27-s − 0.0320·29-s − 1.28·31-s + 0.506·33-s − 0.789·37-s − 1.70·39-s − 0.156·41-s + 0.128·43-s + 0.00335·45-s − 0.679·47-s − 0.920·51-s + 0.613·53-s + 0.771·55-s − 0.888·57-s + 0.440·59-s − 0.0823·61-s − 2.59·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + 6.13T + 13T^{2} \) |
| 17 | \( 1 + 3.78T + 17T^{2} \) |
| 19 | \( 1 + 3.86T + 19T^{2} \) |
| 23 | \( 1 + 5.49T + 23T^{2} \) |
| 29 | \( 1 + 0.172T + 29T^{2} \) |
| 31 | \( 1 + 7.15T + 31T^{2} \) |
| 37 | \( 1 + 4.80T + 37T^{2} \) |
| 43 | \( 1 - 0.845T + 43T^{2} \) |
| 47 | \( 1 + 4.65T + 47T^{2} \) |
| 53 | \( 1 - 4.46T + 53T^{2} \) |
| 59 | \( 1 - 3.38T + 59T^{2} \) |
| 61 | \( 1 + 0.643T + 61T^{2} \) |
| 67 | \( 1 + 2.25T + 67T^{2} \) |
| 71 | \( 1 + 4.77T + 71T^{2} \) |
| 73 | \( 1 + 9.42T + 73T^{2} \) |
| 79 | \( 1 - 3.64T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 6.83T + 89T^{2} \) |
| 97 | \( 1 - 0.615T + 97T^{2} \) |
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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
−7.47946166819086310064142933992, −6.81882169286761274699387092615, −6.11123930489371413728325932336, −5.45231683125993683241343020509, −4.67223861527467539349406156426, −3.82715898479522876660396089640, −2.84481798507939631837339839205, −2.06266635010183292653945975867, −1.90384634501964224949512765236, 0,
1.90384634501964224949512765236, 2.06266635010183292653945975867, 2.84481798507939631837339839205, 3.82715898479522876660396089640, 4.67223861527467539349406156426, 5.45231683125993683241343020509, 6.11123930489371413728325932336, 6.81882169286761274699387092615, 7.47946166819086310064142933992
