| L(s) = 1 | + i·3-s − 3.80i·7-s − 9-s + 0.190·11-s + 1.67i·13-s + 4.60i·17-s + 2.64·19-s + 3.80·21-s − 6.35i·23-s − i·27-s − 2.52·29-s + 3.74·31-s + 0.190i·33-s − 11.8i·37-s − 1.67·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.44i·7-s − 0.333·9-s + 0.0575·11-s + 0.465i·13-s + 1.11i·17-s + 0.605·19-s + 0.831·21-s − 1.32i·23-s − 0.192i·27-s − 0.467·29-s + 0.673·31-s + 0.0332i·33-s − 1.95i·37-s − 0.268·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.335924956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.335924956\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.80iT - 7T^{2} \) |
| 11 | \( 1 - 0.190T + 11T^{2} \) |
| 13 | \( 1 - 1.67iT - 13T^{2} \) |
| 17 | \( 1 - 4.60iT - 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 + 6.35iT - 23T^{2} \) |
| 29 | \( 1 + 2.52T + 29T^{2} \) |
| 31 | \( 1 - 3.74T + 31T^{2} \) |
| 37 | \( 1 + 11.8iT - 37T^{2} \) |
| 41 | \( 1 + 7.18T + 41T^{2} \) |
| 43 | \( 1 - 9.22iT - 43T^{2} \) |
| 47 | \( 1 + 4.54iT - 47T^{2} \) |
| 53 | \( 1 + 9.43iT - 53T^{2} \) |
| 59 | \( 1 + 7.14T + 59T^{2} \) |
| 61 | \( 1 - 9.53T + 61T^{2} \) |
| 67 | \( 1 - 6.05iT - 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 5.21iT - 73T^{2} \) |
| 79 | \( 1 - 3.11T + 79T^{2} \) |
| 83 | \( 1 + 4.67iT - 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 6.69iT - 97T^{2} \) |
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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
−7.81092000123072245614382558776, −6.88887188795939478764729768767, −6.50905611679394731370487170419, −5.52472943484510897157715265630, −4.75578628723749979564006102376, −3.96512541005836230626356229202, −3.71157715181248476771713640773, −2.53915896168930678987325504174, −1.44886142480943587000779761561, −0.34475481504483147196710566370,
1.08724753000852163720313684568, 2.06354907285601388962472426806, 2.88108527394826991982009540288, 3.42739695689099902720153007678, 4.80065056930366103995409272240, 5.35717631753086472082486410955, 5.89414019700048248241031576852, 6.70411505182387200303342358974, 7.37108024555517857215008606737, 8.094622567346778550310640845759
