L(s) = 1 | + (1.36 − 1.36i)2-s − 2.73i·4-s + (0.707 + 0.707i)7-s + (−2.36 − 2.36i)8-s − 0.517i·11-s + 1.93·14-s − 3.73·16-s + (−0.707 − 0.707i)22-s + (0.366 + 0.366i)23-s + (1.93 − 1.93i)28-s − 1.93·29-s + (−2.73 + 2.73i)32-s + (0.707 + 0.707i)37-s + (−0.707 + 0.707i)43-s − 1.41·44-s + ⋯ |
L(s) = 1 | + (1.36 − 1.36i)2-s − 2.73i·4-s + (0.707 + 0.707i)7-s + (−2.36 − 2.36i)8-s − 0.517i·11-s + 1.93·14-s − 3.73·16-s + (−0.707 − 0.707i)22-s + (0.366 + 0.366i)23-s + (1.93 − 1.93i)28-s − 1.93·29-s + (−2.73 + 2.73i)32-s + (0.707 + 0.707i)37-s + (−0.707 + 0.707i)43-s − 1.41·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.268912229\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.268912229\) |
\(L(1)\) |
|
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 | \( 1 + (-0.707 - 0.707i)T \) |
good | 2 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 11 | \( 1 + 0.517iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 29 | \( 1 + 1.93T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1 - i)T + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 71 | \( 1 + 1.93iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.513853471459289408040534317297, −8.908736332295660907067682774952, −7.72383849448969005786842312199, −6.43259580932900426961468598261, −5.64085880659797477565570340569, −5.08567586774935185413730278234, −4.16110148198179180010443735116, −3.26222762422419485066861486374, −2.35368353886260169942350334246, −1.38043885555316304327893211942,
2.22215545205281621639615463225, 3.60628599443616333668334704571, 4.20109767892290615997803522881, 5.08023500784039180141250674472, 5.67263000643270979491281114968, 6.79566737916180409757166308985, 7.24524159390452836084973253421, 7.971603878632795302280240432614, 8.684305056269299596251592458338, 9.740726624498240996135362841153