L(s) = 1 | + (−0.707 − 0.707i)3-s − 5-s + 1.41i·7-s + 1.00i·9-s + (0.707 + 0.707i)15-s + (1.00 − 1.00i)21-s − 1.41·23-s + 25-s + (0.707 − 0.707i)27-s − 1.41i·35-s − 2i·41-s − 1.41·43-s − 1.00i·45-s − 1.41·47-s − 1.00·49-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)3-s − 5-s + 1.41i·7-s + 1.00i·9-s + (0.707 + 0.707i)15-s + (1.00 − 1.00i)21-s − 1.41·23-s + 25-s + (0.707 − 0.707i)27-s − 1.41i·35-s − 2i·41-s − 1.41·43-s − 1.00i·45-s − 1.41·47-s − 1.00·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01237978921\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01237978921\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 - 1.41iT - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.41T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + 2iT - T^{2} \) |
| 43 | \( 1 + 1.41T + T^{2} \) |
| 47 | \( 1 + 1.41T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.41T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 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
−8.697286999566838953986846959090, −8.305461805018408531215567204887, −7.57192710449211278444612557541, −6.81324003973230557064941629380, −6.06118736793739334712054466473, −5.40954698290835439253901618813, −4.67052556955220412421916382772, −3.63240466849811611118150125263, −2.57070507008022208528793041421, −1.68649776864335353221787410115,
0.008150018827572329029354412215, 1.31464935370800166507967300460, 3.15671669102598803735436745692, 3.81356137558899594991905410199, 4.45057210495473343033195865516, 5.01200135443536793589265335769, 6.22461103655919537040496537999, 6.73260515218850031697542775123, 7.64082764386813457700430676262, 8.121505881133523541920206691540