| L(s) = 1 | − 0.765·3-s + 1.41·5-s + 1.84·7-s − 0.414·9-s − 1.84·11-s − 1.08·15-s + 0.765·19-s − 1.41·21-s + 1.00·25-s + 1.08·27-s + 1.41·33-s + 2.61·35-s − 1.41·37-s − 41-s − 0.585·45-s + 0.765·47-s + 2.41·49-s − 2.61·55-s − 0.585·57-s − 0.765·63-s − 1.84·67-s − 0.765·71-s − 1.41·73-s − 0.765·75-s − 3.41·77-s + 0.765·79-s − 0.414·81-s + ⋯ |
| L(s) = 1 | − 0.765·3-s + 1.41·5-s + 1.84·7-s − 0.414·9-s − 1.84·11-s − 1.08·15-s + 0.765·19-s − 1.41·21-s + 1.00·25-s + 1.08·27-s + 1.41·33-s + 2.61·35-s − 1.41·37-s − 41-s − 0.585·45-s + 0.765·47-s + 2.41·49-s − 2.61·55-s − 0.585·57-s − 0.765·63-s − 1.84·67-s − 0.765·71-s − 1.41·73-s − 0.765·75-s − 3.41·77-s + 0.765·79-s − 0.414·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9702259687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9702259687\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 0.765T + T^{2} \) |
| 5 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - 1.84T + T^{2} \) |
| 11 | \( 1 + 1.84T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 0.765T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 0.765T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.84T + T^{2} \) |
| 71 | \( 1 + 0.765T + T^{2} \) |
| 73 | \( 1 + 1.41T + T^{2} \) |
| 79 | \( 1 - 0.765T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.54913124405329960785569175695, −10.34548027905648221112671227930, −8.965496522152418643214323052203, −8.153793755702559167855268379400, −7.26078090693194384009937625857, −5.86544379514963196660341740677, −5.32112748483815526770213425369, −4.84424762854838074947685567446, −2.72422454665248206811974865166, −1.64762374301951339998782882748,
1.64762374301951339998782882748, 2.72422454665248206811974865166, 4.84424762854838074947685567446, 5.32112748483815526770213425369, 5.86544379514963196660341740677, 7.26078090693194384009937625857, 8.153793755702559167855268379400, 8.965496522152418643214323052203, 10.34548027905648221112671227930, 10.54913124405329960785569175695
