L(s) = 1 | + (0.965 + 0.258i)3-s + (0.965 − 0.258i)7-s + (0.866 + 0.499i)9-s + 21-s + (−0.448 + 1.67i)23-s + (0.707 + 0.707i)27-s + (−0.866 + 1.5i)29-s + (1.5 − 0.866i)41-s + (−0.517 − 1.93i)43-s + (−0.448 − 1.67i)47-s + (0.5 − 0.866i)61-s + (0.965 + 0.258i)63-s + (0.258 − 0.965i)67-s + (−0.866 + 1.50i)69-s + (0.500 + 0.866i)81-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)3-s + (0.965 − 0.258i)7-s + (0.866 + 0.499i)9-s + 21-s + (−0.448 + 1.67i)23-s + (0.707 + 0.707i)27-s + (−0.866 + 1.5i)29-s + (1.5 − 0.866i)41-s + (−0.517 − 1.93i)43-s + (−0.448 − 1.67i)47-s + (0.5 − 0.866i)61-s + (0.965 + 0.258i)63-s + (0.258 − 0.965i)67-s + (−0.866 + 1.50i)69-s + (0.500 + 0.866i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.041735978\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.041735978\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.965 - 0.258i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.517 + 1.93i)T + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 - 1.73T + T^{2} \) |
| 97 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
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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
−8.799527111852368285877547178167, −8.017540698127703811382767468588, −7.47150701975341430642304746948, −6.85636535378470935821228057564, −5.49051909633618327065897557243, −5.02362173202158865165236819361, −3.91184943482388378554083384401, −3.48690730538552668443091002794, −2.19778982260867189939822959899, −1.50489135311635730710658114436,
1.26407727617359845224508479476, 2.28164076821280251317869923389, 2.92937821784676788104727804471, 4.24732632923913498142731094552, 4.53585094136889245368421436789, 5.81371677543938001640669424844, 6.49800145272652026127835540317, 7.45862795279270760424997080492, 8.129679974408588506510377941400, 8.391223462774185736284279116282