| L(s) = 1 | − 3·3-s + 5·5-s + 14.0·7-s + 9·9-s − 50.8·11-s + 38.0·13-s − 15·15-s − 2.85·17-s + 31.7·19-s − 42.0·21-s − 23·23-s + 25·25-s − 27·27-s + 258.·29-s − 165.·31-s + 152.·33-s + 70.1·35-s − 153.·37-s − 114.·39-s + 494.·41-s + 29.8·43-s + 45·45-s − 130.·47-s − 146.·49-s + 8.57·51-s + 177.·53-s − 254.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.757·7-s + 0.333·9-s − 1.39·11-s + 0.811·13-s − 0.258·15-s − 0.0407·17-s + 0.383·19-s − 0.437·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s + 1.65·29-s − 0.957·31-s + 0.805·33-s + 0.338·35-s − 0.680·37-s − 0.468·39-s + 1.88·41-s + 0.105·43-s + 0.149·45-s − 0.405·47-s − 0.426·49-s + 0.0235·51-s + 0.461·53-s − 0.623·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.957315795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.957315795\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 7 | \( 1 - 14.0T + 343T^{2} \) |
| 11 | \( 1 + 50.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.85T + 4.91e3T^{2} \) |
| 19 | \( 1 - 31.7T + 6.85e3T^{2} \) |
| 29 | \( 1 - 258.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 153.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 494.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 29.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 130.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 177.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 559.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 508.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 104.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 44.3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 190.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 412.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 147.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 464.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 279.T + 9.12e5T^{2} \) |
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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
−9.231476621298779985828402151806, −8.298655074004048489297161375711, −7.67257932547405558898360254187, −6.67742732661564056951396036496, −5.73712024451791097129764676737, −5.17353725322476013537746603730, −4.28894800646861019317902321918, −2.96824186004345437786111571591, −1.86617988511706602459544623942, −0.72916615557712116441162657346,
0.72916615557712116441162657346, 1.86617988511706602459544623942, 2.96824186004345437786111571591, 4.28894800646861019317902321918, 5.17353725322476013537746603730, 5.73712024451791097129764676737, 6.67742732661564056951396036496, 7.67257932547405558898360254187, 8.298655074004048489297161375711, 9.231476621298779985828402151806
