| L(s) = 1 | − 2-s − 1.41·3-s + 4-s + 1.41·6-s + 7-s − 8-s + 1.00·9-s − 1.41·12-s + 1.41·13-s − 14-s + 16-s − 1.00·18-s − 1.41·19-s − 1.41·21-s + 1.41·24-s − 1.41·26-s + 28-s − 32-s + 1.00·36-s + 1.41·38-s − 2.00·39-s + 1.41·42-s − 1.41·48-s + 49-s + 1.41·52-s − 56-s + 2.00·57-s + ⋯ |
| L(s) = 1 | − 2-s − 1.41·3-s + 4-s + 1.41·6-s + 7-s − 8-s + 1.00·9-s − 1.41·12-s + 1.41·13-s − 14-s + 16-s − 1.00·18-s − 1.41·19-s − 1.41·21-s + 1.41·24-s − 1.41·26-s + 28-s − 32-s + 1.00·36-s + 1.41·38-s − 2.00·39-s + 1.41·42-s − 1.41·48-s + 49-s + 1.41·52-s − 56-s + 2.00·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4990776815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4990776815\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.41T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 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
−10.00145615138226631232542030836, −8.724630634173229318593966903615, −8.375555927139315546491184104959, −7.29874567383983547375946247741, −6.41993972291292251686613973530, −5.89255768538943068111472731722, −4.95822600815069049158835218240, −3.82044374204711435348358272558, −2.11881067759400590285768240661, −0.979259789588622952788754477622,
0.979259789588622952788754477622, 2.11881067759400590285768240661, 3.82044374204711435348358272558, 4.95822600815069049158835218240, 5.89255768538943068111472731722, 6.41993972291292251686613973530, 7.29874567383983547375946247741, 8.375555927139315546491184104959, 8.724630634173229318593966903615, 10.00145615138226631232542030836
