| L(s) = 1 | − 0.648·3-s − 1.99·5-s + 7-s − 2.57·9-s + 11-s − 13-s + 1.29·15-s + 4.48·17-s + 3.80·19-s − 0.648·21-s + 6.17·23-s − 1.01·25-s + 3.61·27-s − 9.03·29-s − 8.94·31-s − 0.648·33-s − 1.99·35-s − 0.194·37-s + 0.648·39-s − 7.67·41-s + 5.29·43-s + 5.15·45-s − 0.987·47-s + 49-s − 2.90·51-s − 12.4·53-s − 1.99·55-s + ⋯ |
| L(s) = 1 | − 0.374·3-s − 0.892·5-s + 0.377·7-s − 0.859·9-s + 0.301·11-s − 0.277·13-s + 0.334·15-s + 1.08·17-s + 0.872·19-s − 0.141·21-s + 1.28·23-s − 0.202·25-s + 0.695·27-s − 1.67·29-s − 1.60·31-s − 0.112·33-s − 0.337·35-s − 0.0319·37-s + 0.103·39-s − 1.19·41-s + 0.808·43-s + 0.767·45-s − 0.144·47-s + 0.142·49-s − 0.406·51-s − 1.71·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.116413377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.116413377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.648T + 3T^{2} \) |
| 5 | \( 1 + 1.99T + 5T^{2} \) |
| 17 | \( 1 - 4.48T + 17T^{2} \) |
| 19 | \( 1 - 3.80T + 19T^{2} \) |
| 23 | \( 1 - 6.17T + 23T^{2} \) |
| 29 | \( 1 + 9.03T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 + 0.194T + 37T^{2} \) |
| 41 | \( 1 + 7.67T + 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 + 0.987T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 7.62T + 59T^{2} \) |
| 61 | \( 1 - 2.48T + 61T^{2} \) |
| 67 | \( 1 - 15.9T + 67T^{2} \) |
| 71 | \( 1 - 5.07T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 5.85T + 89T^{2} \) |
| 97 | \( 1 - 7.77T + 97T^{2} \) |
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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
−8.365011735709010813768431676323, −7.58115188212123992938489166429, −7.25828303619722687284167613332, −6.13612085782820047305118868150, −5.35677464844296006221323189208, −4.89191059899987619208706502195, −3.62008033563049237010500692424, −3.27611324880195826026044716465, −1.87865606246870239840851066363, −0.61368916717975188123239548961,
0.61368916717975188123239548961, 1.87865606246870239840851066363, 3.27611324880195826026044716465, 3.62008033563049237010500692424, 4.89191059899987619208706502195, 5.35677464844296006221323189208, 6.13612085782820047305118868150, 7.25828303619722687284167613332, 7.58115188212123992938489166429, 8.365011735709010813768431676323
