| L(s) = 1 | − 1.76·3-s − 7-s + 0.103·9-s − 0.626·11-s − 5.49·13-s − 0.896·17-s + 6.38·19-s + 1.76·21-s + 3.72·23-s + 5.10·27-s − 7.87·29-s + 7.52·31-s + 1.10·33-s − 6·37-s + 9.67·39-s + 7.72·41-s − 1.72·43-s − 5.87·47-s + 49-s + 1.57·51-s + 6.77·53-s − 11.2·57-s − 0.593·59-s + 7.13·61-s − 0.103·63-s + 5.79·67-s − 6.56·69-s + ⋯ |
| L(s) = 1 | − 1.01·3-s − 0.377·7-s + 0.0343·9-s − 0.188·11-s − 1.52·13-s − 0.217·17-s + 1.46·19-s + 0.384·21-s + 0.777·23-s + 0.982·27-s − 1.46·29-s + 1.35·31-s + 0.192·33-s − 0.986·37-s + 1.54·39-s + 1.20·41-s − 0.263·43-s − 0.857·47-s + 0.142·49-s + 0.221·51-s + 0.930·53-s − 1.49·57-s − 0.0773·59-s + 0.913·61-s − 0.0129·63-s + 0.707·67-s − 0.790·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8214227965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8214227965\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 1.76T + 3T^{2} \) |
| 11 | \( 1 + 0.626T + 11T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 17 | \( 1 + 0.896T + 17T^{2} \) |
| 19 | \( 1 - 6.38T + 19T^{2} \) |
| 23 | \( 1 - 3.72T + 23T^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 31 | \( 1 - 7.52T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 7.72T + 41T^{2} \) |
| 43 | \( 1 + 1.72T + 43T^{2} \) |
| 47 | \( 1 + 5.87T + 47T^{2} \) |
| 53 | \( 1 - 6.77T + 53T^{2} \) |
| 59 | \( 1 + 0.593T + 59T^{2} \) |
| 61 | \( 1 - 7.13T + 61T^{2} \) |
| 67 | \( 1 - 5.79T + 67T^{2} \) |
| 71 | \( 1 - 5.52T + 71T^{2} \) |
| 73 | \( 1 - 3.72T + 73T^{2} \) |
| 79 | \( 1 + 5.67T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−9.708516420451342416624553156024, −8.918052796131110514510744961704, −7.71845745435832364682029767468, −7.08444132262771253789531127816, −6.22969908981643172555909007877, −5.26522351164895423757386557354, −4.88323502689938343491147495418, −3.45591462771770814431260870441, −2.40096966435580942129278789581, −0.66632876187762090243030286651,
0.66632876187762090243030286651, 2.40096966435580942129278789581, 3.45591462771770814431260870441, 4.88323502689938343491147495418, 5.26522351164895423757386557354, 6.22969908981643172555909007877, 7.08444132262771253789531127816, 7.71845745435832364682029767468, 8.918052796131110514510744961704, 9.708516420451342416624553156024
