L(s) = 1 | + (−0.707 − 0.707i)3-s + (0.707 + 0.707i)5-s + (0.707 − 0.707i)7-s + 1.00i·9-s − 1.00i·15-s + (−1.41 − 1.41i)17-s + 1.41·19-s − 1.00·21-s + (−1 − i)23-s + 1.00i·25-s + (0.707 − 0.707i)27-s − 1.41i·31-s + 1.00·35-s + (1 + i)37-s + 1.41·41-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)3-s + (0.707 + 0.707i)5-s + (0.707 − 0.707i)7-s + 1.00i·9-s − 1.00i·15-s + (−1.41 − 1.41i)17-s + 1.41·19-s − 1.00·21-s + (−1 − i)23-s + 1.00i·25-s + (0.707 − 0.707i)27-s − 1.41i·31-s + 1.00·35-s + (1 + i)37-s + 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.184420400\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.184420400\) |
\(L(1)\) |
|
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 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 19 | \( 1 - 1.41T + T^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + 1.41iT - T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - 2iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.551133033319785223672851504829, −7.56219344637655923753388254691, −7.27642380648245057903603499349, −6.43509908595508887185457534471, −5.82841111855943396181191059710, −4.91512787662359043555441658643, −4.25281533292078090539591123819, −2.75705335414895335845842959015, −2.09920709404672417074003357108, −0.862356645613774447445871601511,
1.30872187858092348319206247817, 2.26421038654408395243353952154, 3.63272617502131183661179476308, 4.51275969385245733385021408067, 5.12958166396942435341787939649, 5.87637375798843160316295720174, 6.24667371624717261620288880249, 7.50064139810412005641180398644, 8.378159477520016652842088822602, 9.114165196931640697342330676187