| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.207780910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.207780910\) |
| \(L(1)\) |
|
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
−9.539879422217702921626735452615, −8.978059875023467949933676766683, −8.140939753357669301451261887400, −7.60278033682101438168505721462, −7.11327451894400425567814646014, −5.67631954318862370430598278022, −4.61969348407239127380924462246, −3.26516048459389391352190780139, −2.49304226006731570545255365727, −1.53885203955073937888686704228,
1.53885203955073937888686704228, 2.49304226006731570545255365727, 3.26516048459389391352190780139, 4.61969348407239127380924462246, 5.67631954318862370430598278022, 7.11327451894400425567814646014, 7.60278033682101438168505721462, 8.140939753357669301451261887400, 8.978059875023467949933676766683, 9.539879422217702921626735452615
