L(s) = 1 | − 3-s − 1.41·7-s + 9-s − 1.41·13-s + 1.41·21-s + 25-s − 27-s + 1.41·31-s + 1.41·37-s + 1.41·39-s + 1.00·49-s − 1.41·61-s − 1.41·63-s + 2·67-s − 75-s + 1.41·79-s + 81-s + 2.00·91-s − 1.41·93-s + 1.41·103-s + 1.41·109-s − 1.41·111-s − 1.41·117-s + ⋯ |
L(s) = 1 | − 3-s − 1.41·7-s + 9-s − 1.41·13-s + 1.41·21-s + 25-s − 27-s + 1.41·31-s + 1.41·37-s + 1.41·39-s + 1.00·49-s − 1.41·61-s − 1.41·63-s + 2·67-s − 75-s + 1.41·79-s + 81-s + 2.00·91-s − 1.41·93-s + 1.41·103-s + 1.41·109-s − 1.41·111-s − 1.41·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6346548837\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6346548837\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
good | 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 - 1.41T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 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
−9.164942597516941157223737005527, −7.971611243004966207134379706379, −7.16402196395121179170066450735, −6.55971558286211727137471928660, −6.00568393132403235936220870419, −5.02556452087937922801878934744, −4.41318797135101719387637790001, −3.27882373347769265471598047941, −2.38014133894110123122995505644, −0.72738376297307692097239513453,
0.72738376297307692097239513453, 2.38014133894110123122995505644, 3.27882373347769265471598047941, 4.41318797135101719387637790001, 5.02556452087937922801878934744, 6.00568393132403235936220870419, 6.55971558286211727137471928660, 7.16402196395121179170066450735, 7.971611243004966207134379706379, 9.164942597516941157223737005527