L(s) = 1 | + 2.67i·2-s − 5.13·4-s + 5-s + (1.64 − 2.06i)7-s − 8.37i·8-s + 2.67i·10-s − 4.55i·11-s + 4.75i·13-s + (5.52 + 4.40i)14-s + 12.0·16-s + 2.70·17-s + 6.55i·19-s − 5.13·20-s + 12.1·22-s + 6.74i·23-s + ⋯ |
L(s) = 1 | + 1.88i·2-s − 2.56·4-s + 0.447·5-s + (0.623 − 0.782i)7-s − 2.95i·8-s + 0.844i·10-s − 1.37i·11-s + 1.31i·13-s + (1.47 + 1.17i)14-s + 3.02·16-s + 0.656·17-s + 1.50i·19-s − 1.14·20-s + 2.59·22-s + 1.40i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.782 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.504739 + 1.44371i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.504739 + 1.44371i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-1.64 + 2.06i)T \) |
good | 2 | \( 1 - 2.67iT - 2T^{2} \) |
| 11 | \( 1 + 4.55iT - 11T^{2} \) |
| 13 | \( 1 - 4.75iT - 13T^{2} \) |
| 17 | \( 1 - 2.70T + 17T^{2} \) |
| 19 | \( 1 - 6.55iT - 19T^{2} \) |
| 23 | \( 1 - 6.74iT - 23T^{2} \) |
| 29 | \( 1 + 3.19iT - 29T^{2} \) |
| 31 | \( 1 - 7.05iT - 31T^{2} \) |
| 37 | \( 1 - 8.66T + 37T^{2} \) |
| 41 | \( 1 - 4.62T + 41T^{2} \) |
| 43 | \( 1 - 3.29T + 43T^{2} \) |
| 47 | \( 1 - 9.69T + 47T^{2} \) |
| 53 | \( 1 + 3.37iT - 53T^{2} \) |
| 59 | \( 1 - 2.55T + 59T^{2} \) |
| 61 | \( 1 - 4.84iT - 61T^{2} \) |
| 67 | \( 1 + 2.18T + 67T^{2} \) |
| 71 | \( 1 - 1.61iT - 71T^{2} \) |
| 73 | \( 1 + 9.66iT - 73T^{2} \) |
| 79 | \( 1 + 1.73T + 79T^{2} \) |
| 83 | \( 1 + 5.20T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 0.365iT - 97T^{2} \) |
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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
−9.993642846410115729462292553934, −9.252225511758506634370078168372, −8.418618354879227060480150130098, −7.74414268185903987573261712518, −7.03940343193959755445704675575, −5.99041450041697681917542525343, −5.60436851128393459691510948175, −4.41727743244566018791337499185, −3.64085537218189063127317800137, −1.20984877204190532941386729791,
0.919108864825219249082512853761, 2.35610277170470301673851307945, 2.70743674327382347814335429625, 4.27409275562576219358860376337, 4.94372108428768793603484892018, 5.82327801896446611197440728653, 7.49824887161161114201065468555, 8.453819530705059089683295873185, 9.264155134014800595064906929306, 9.855209591516055417363993525583