| L(s) = 1 | + 0.858·2-s − 3-s − 1.26·4-s − 0.858·6-s + 3.88·7-s − 2.80·8-s + 9-s − 1.39·11-s + 1.26·12-s + 3.36·13-s + 3.33·14-s + 0.118·16-s + 3.11·17-s + 0.858·18-s − 2.70·19-s − 3.88·21-s − 1.20·22-s − 6.43·23-s + 2.80·24-s + 2.88·26-s − 27-s − 4.89·28-s + 8.26·29-s − 6.34·31-s + 5.70·32-s + 1.39·33-s + 2.67·34-s + ⋯ |
| L(s) = 1 | + 0.607·2-s − 0.577·3-s − 0.631·4-s − 0.350·6-s + 1.46·7-s − 0.990·8-s + 0.333·9-s − 0.421·11-s + 0.364·12-s + 0.932·13-s + 0.890·14-s + 0.0296·16-s + 0.755·17-s + 0.202·18-s − 0.620·19-s − 0.846·21-s − 0.256·22-s − 1.34·23-s + 0.571·24-s + 0.566·26-s − 0.192·27-s − 0.925·28-s + 1.53·29-s − 1.13·31-s + 1.00·32-s + 0.243·33-s + 0.458·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.827929627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.827929627\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.858T + 2T^{2} \) |
| 7 | \( 1 - 3.88T + 7T^{2} \) |
| 11 | \( 1 + 1.39T + 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 - 3.11T + 17T^{2} \) |
| 19 | \( 1 + 2.70T + 19T^{2} \) |
| 23 | \( 1 + 6.43T + 23T^{2} \) |
| 29 | \( 1 - 8.26T + 29T^{2} \) |
| 31 | \( 1 + 6.34T + 31T^{2} \) |
| 37 | \( 1 + 7.49T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 3.39T + 43T^{2} \) |
| 47 | \( 1 - 8.38T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 7.64T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 3.54T + 67T^{2} \) |
| 71 | \( 1 - 1.18T + 71T^{2} \) |
| 73 | \( 1 - 2.39T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 1.40T + 83T^{2} \) |
| 89 | \( 1 - 4.62T + 89T^{2} \) |
| 97 | \( 1 + 1.51T + 97T^{2} \) |
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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
−9.094010807246931879013824786894, −8.346114523438037563986767794839, −7.84519572193168160854220809747, −6.64853578949490057786815805660, −5.62169363176808210536296608505, −5.31008352212082031948498031809, −4.31075056927060029191835508244, −3.75352544513448775842440088679, −2.22916478683311507921517424111, −0.898779809436040667508083546091,
0.898779809436040667508083546091, 2.22916478683311507921517424111, 3.75352544513448775842440088679, 4.31075056927060029191835508244, 5.31008352212082031948498031809, 5.62169363176808210536296608505, 6.64853578949490057786815805660, 7.84519572193168160854220809747, 8.346114523438037563986767794839, 9.094010807246931879013824786894
