| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2127473825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2127473825\) |
| \(L(1)\) |
|
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
−9.038597090131439831089465824093, −8.127109184562639766440799811719, −7.53018040152150181318281747098, −6.76808140230392727389084236282, −5.93955290880403532173534392679, −5.26297605638245412241856276540, −4.30221010155252967085880246593, −3.18575805913176957651860779903, −2.85447297829314229848466532601, −0.40869646541742863523646276767,
0.40869646541742863523646276767, 2.85447297829314229848466532601, 3.18575805913176957651860779903, 4.30221010155252967085880246593, 5.26297605638245412241856276540, 5.93955290880403532173534392679, 6.76808140230392727389084236282, 7.53018040152150181318281747098, 8.127109184562639766440799811719, 9.038597090131439831089465824093
