| L(s) = 1 | − 9.59·3-s − 0.0570·5-s − 24.6·7-s + 65.1·9-s − 43.3·11-s − 36.6·13-s + 0.547·15-s + 65.8·17-s + 19·19-s + 236.·21-s + 110.·23-s − 124.·25-s − 366.·27-s + 91.1·29-s − 148.·31-s + 415.·33-s + 1.40·35-s + 89.4·37-s + 352.·39-s + 414.·41-s − 104.·43-s − 3.71·45-s + 602.·47-s + 262.·49-s − 631.·51-s − 187.·53-s + 2.46·55-s + ⋯ |
| L(s) = 1 | − 1.84·3-s − 0.00509·5-s − 1.32·7-s + 2.41·9-s − 1.18·11-s − 0.782·13-s + 0.00941·15-s + 0.939·17-s + 0.229·19-s + 2.45·21-s + 0.998·23-s − 0.999·25-s − 2.60·27-s + 0.583·29-s − 0.858·31-s + 2.19·33-s + 0.00677·35-s + 0.397·37-s + 1.44·39-s + 1.57·41-s − 0.369·43-s − 0.0123·45-s + 1.87·47-s + 0.765·49-s − 1.73·51-s − 0.484·53-s + 0.00605·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 + 9.59T + 27T^{2} \) |
| 5 | \( 1 + 0.0570T + 125T^{2} \) |
| 7 | \( 1 + 24.6T + 343T^{2} \) |
| 11 | \( 1 + 43.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 36.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 65.8T + 4.91e3T^{2} \) |
| 23 | \( 1 - 110.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 91.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 148.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 89.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 104.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 602.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 187.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 283.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 539.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 730.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 707.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 989.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 897.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 388.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 17.3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 721.T + 9.12e5T^{2} \) |
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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
−9.399374109916074936967932482797, −7.74002107469795659897578118378, −7.15822458995909542674357404223, −6.27698636643782193598983065512, −5.57194763642851550367909909107, −5.00548801607587185353994854069, −3.82922524355170859501606146299, −2.57884142974014258804286648350, −0.860384655928935524870333912056, 0,
0.860384655928935524870333912056, 2.57884142974014258804286648350, 3.82922524355170859501606146299, 5.00548801607587185353994854069, 5.57194763642851550367909909107, 6.27698636643782193598983065512, 7.15822458995909542674357404223, 7.74002107469795659897578118378, 9.399374109916074936967932482797
