| L(s) = 1 | + 2.18·2-s + 3-s + 2.79·4-s + 2.18·6-s − 1.73·7-s + 1.73·8-s − 2·9-s − 2.64·11-s + 2.79·12-s − 3.79·14-s − 1.79·16-s − 4.58·17-s − 4.37·18-s − 1.73·19-s − 1.73·21-s − 5.79·22-s + 4.58·23-s + 1.73·24-s − 5·27-s − 4.83·28-s − 4.58·29-s + 9.66·31-s − 7.38·32-s − 2.64·33-s − 10.0·34-s − 5.58·36-s − 7.93·37-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 0.577·3-s + 1.39·4-s + 0.893·6-s − 0.654·7-s + 0.612·8-s − 0.666·9-s − 0.797·11-s + 0.805·12-s − 1.01·14-s − 0.447·16-s − 1.11·17-s − 1.03·18-s − 0.397·19-s − 0.377·21-s − 1.23·22-s + 0.955·23-s + 0.353·24-s − 0.962·27-s − 0.913·28-s − 0.850·29-s + 1.73·31-s − 1.30·32-s − 0.460·33-s − 1.72·34-s − 0.930·36-s − 1.30·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.18T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 + 1.73T + 19T^{2} \) |
| 23 | \( 1 - 4.58T + 23T^{2} \) |
| 29 | \( 1 + 4.58T + 29T^{2} \) |
| 31 | \( 1 - 9.66T + 31T^{2} \) |
| 37 | \( 1 + 7.93T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 - 8.75T + 47T^{2} \) |
| 53 | \( 1 + 1.58T + 53T^{2} \) |
| 59 | \( 1 - 3.36T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 - 3.55T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 4.28T + 89T^{2} \) |
| 97 | \( 1 + 4.47T + 97T^{2} \) |
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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
−7.943971645732736292516152681141, −7.02020598317645414473120822124, −6.40653274144253591804910826330, −5.70309751905196841787709033739, −4.95859399430338701121897177409, −4.25202024765824550805103540326, −3.29426784704222992747145366046, −2.84520654844099799195705295599, −2.06467378330474885516058628763, 0,
2.06467378330474885516058628763, 2.84520654844099799195705295599, 3.29426784704222992747145366046, 4.25202024765824550805103540326, 4.95859399430338701121897177409, 5.70309751905196841787709033739, 6.40653274144253591804910826330, 7.02020598317645414473120822124, 7.943971645732736292516152681141
