L(s) = 1 | + (0.866 − 0.5i)4-s + (0.258 − 0.965i)7-s + (−1.22 − 1.22i)13-s + (0.499 − 0.866i)16-s + (−0.866 − 0.5i)19-s + (−0.258 − 0.965i)28-s + (1 + 1.73i)31-s + (0.448 + 1.67i)37-s + (−0.866 − 0.499i)49-s + (−1.67 − 0.448i)52-s + (0.5 − 0.866i)61-s − 0.999i·64-s + (1.67 + 0.448i)67-s + (−0.448 + 1.67i)73-s − 0.999·76-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)4-s + (0.258 − 0.965i)7-s + (−1.22 − 1.22i)13-s + (0.499 − 0.866i)16-s + (−0.866 − 0.5i)19-s + (−0.258 − 0.965i)28-s + (1 + 1.73i)31-s + (0.448 + 1.67i)37-s + (−0.866 − 0.499i)49-s + (−1.67 − 0.448i)52-s + (0.5 − 0.866i)61-s − 0.999i·64-s + (1.67 + 0.448i)67-s + (−0.448 + 1.67i)73-s − 0.999·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.292459688\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.292459688\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.258 + 0.965i)T \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
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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
−9.951592925000635889525354481582, −8.494141219863861204969707475587, −7.78774483164192325258540193149, −6.98180430968360976672507201809, −6.44816496853024155598705855075, −5.25094658998857753458134835169, −4.66501967768058129842898324007, −3.27825115821248863519506582903, −2.39203838478409052843605799611, −1.03032890668133285385043912399,
2.08500688479143862743811439639, 2.43098139386542800302091404075, 3.83311240615520857392924160459, 4.74012550645780998212308089202, 5.89519199970182957966490877131, 6.50283566932256669455131674809, 7.44877139238149787483532861420, 8.059007735899930348440632505708, 8.981799427800578227010804549900, 9.657223612762648477945291476773