L(s) = 1 | + (−0.923 + 0.382i)3-s + (−0.923 + 0.382i)5-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)9-s + (0.541 − 0.541i)13-s + (0.707 − 0.707i)15-s − 1.84i·19-s + (−0.923 − 0.382i)21-s + (−1 − i)23-s + (0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s + (−0.923 − 0.382i)35-s + (−0.292 + 0.707i)39-s + (−0.382 + 0.923i)45-s + 1.00i·49-s + ⋯ |
L(s) = 1 | + (−0.923 + 0.382i)3-s + (−0.923 + 0.382i)5-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)9-s + (0.541 − 0.541i)13-s + (0.707 − 0.707i)15-s − 1.84i·19-s + (−0.923 − 0.382i)21-s + (−1 − i)23-s + (0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s + (−0.923 − 0.382i)35-s + (−0.292 + 0.707i)39-s + (−0.382 + 0.923i)45-s + 1.00i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7822643368\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7822643368\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.923 - 0.382i)T \) |
| 5 | \( 1 + (0.923 - 0.382i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + 1.84iT - T^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - 1.84T + T^{2} \) |
| 61 | \( 1 - 0.765T + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 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
−8.667031603173324102604297460613, −8.106887480379466983372439208106, −7.17355426028736612146142782560, −6.51588057685523025689980920026, −5.71146910231483742461236121147, −4.87441371073937185683998871691, −4.32518644132577334412914234709, −3.34844838304635508756046542305, −2.28981581832743426789709247341, −0.66058366343829626933164759105,
1.07697561750994030661509749579, 1.89008385081340470696412097661, 3.81511949704210866873632560155, 4.02696128023889474325705469580, 5.08804196021603773504475164680, 5.73849125966987744657956432597, 6.65597085047542635699006685518, 7.42087823347574052468484019021, 7.977860098679575395274735440686, 8.493468660835387657422145276203