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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.191722508\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.191722508\) |
\(L(1)\) |
|
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
−8.791995848079769529551641207198, −8.069539001429748004715702400538, −7.56096087748299206719074873641, −6.35042994814027866804031253820, −5.96352728828073566109723017889, −4.88521445422036579731262649182, −4.16771483394908363987549486887, −2.98165141913523582906484344811, −2.02104356333769841329614070332, −1.60027795934095171860718553013,
1.45164395205423187010011638461, 2.41571812206722988649321887569, 3.13208003276914848998773191894, 4.19792346534997651625672872917, 4.92412449807120061149027990378, 5.66438783638597269822277131366, 6.85507619398593051100216084780, 7.36544104610469560203597445596, 8.090588858044885251980701886424, 8.883166406361695719738837841275