| L(s) = 1 | − 0.820·3-s + 5-s + 7-s − 2.32·9-s − 1.23·11-s + 13-s − 0.820·15-s − 0.313·17-s + 1.90·19-s − 0.820·21-s + 7.89·23-s + 25-s + 4.36·27-s + 3.42·29-s + 6.27·31-s + 1.01·33-s + 35-s − 4.05·37-s − 0.820·39-s − 7.79·41-s − 2.87·43-s − 2.32·45-s − 11.5·47-s + 49-s + 0.256·51-s − 4.65·53-s − 1.23·55-s + ⋯ |
| L(s) = 1 | − 0.473·3-s + 0.447·5-s + 0.377·7-s − 0.775·9-s − 0.372·11-s + 0.277·13-s − 0.211·15-s − 0.0759·17-s + 0.437·19-s − 0.178·21-s + 1.64·23-s + 0.200·25-s + 0.840·27-s + 0.636·29-s + 1.12·31-s + 0.176·33-s + 0.169·35-s − 0.666·37-s − 0.131·39-s − 1.21·41-s − 0.438·43-s − 0.346·45-s − 1.69·47-s + 0.142·49-s + 0.0359·51-s − 0.639·53-s − 0.166·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.746756914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.746756914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 0.820T + 3T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 17 | \( 1 + 0.313T + 17T^{2} \) |
| 19 | \( 1 - 1.90T + 19T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 29 | \( 1 - 3.42T + 29T^{2} \) |
| 31 | \( 1 - 6.27T + 31T^{2} \) |
| 37 | \( 1 + 4.05T + 37T^{2} \) |
| 41 | \( 1 + 7.79T + 41T^{2} \) |
| 43 | \( 1 + 2.87T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 4.65T + 53T^{2} \) |
| 59 | \( 1 - 9.50T + 59T^{2} \) |
| 61 | \( 1 + 3.01T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 3.77T + 71T^{2} \) |
| 73 | \( 1 + 5.66T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.121271669434680340707712478493, −6.98021709702041909189405968427, −6.58042037922624343304564455672, −5.72422047871529341914623621194, −5.09212551175002406360213395908, −4.68409337103004063583592414490, −3.34237963645868058478998714104, −2.81697853973648219990274298389, −1.70882663417876192899156545410, −0.69639712482011592391645673401,
0.69639712482011592391645673401, 1.70882663417876192899156545410, 2.81697853973648219990274298389, 3.34237963645868058478998714104, 4.68409337103004063583592414490, 5.09212551175002406360213395908, 5.72422047871529341914623621194, 6.58042037922624343304564455672, 6.98021709702041909189405968427, 8.121271669434680340707712478493
