L(s) = 1 | + 0.356i·3-s + (0.376 + 2.20i)5-s + 2.46i·7-s + 2.87·9-s + 1.61·11-s + 2.62i·13-s + (−0.785 + 0.134i)15-s − 2.58i·17-s − 4.02·19-s − 0.878·21-s − i·23-s + (−4.71 + 1.65i)25-s + 2.09i·27-s + 7.08·29-s − 4.58·31-s + ⋯ |
L(s) = 1 | + 0.205i·3-s + (0.168 + 0.985i)5-s + 0.931i·7-s + 0.957·9-s + 0.487·11-s + 0.728i·13-s + (−0.202 + 0.0346i)15-s − 0.627i·17-s − 0.923·19-s − 0.191·21-s − 0.208i·23-s + (−0.943 + 0.331i)25-s + 0.402i·27-s + 1.31·29-s − 0.823·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05380 + 1.24893i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05380 + 1.24893i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.376 - 2.20i)T \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 - 0.356iT - 3T^{2} \) |
| 7 | \( 1 - 2.46iT - 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 - 2.62iT - 13T^{2} \) |
| 17 | \( 1 + 2.58iT - 17T^{2} \) |
| 19 | \( 1 + 4.02T + 19T^{2} \) |
| 29 | \( 1 - 7.08T + 29T^{2} \) |
| 31 | \( 1 + 4.58T + 31T^{2} \) |
| 37 | \( 1 - 2.96iT - 37T^{2} \) |
| 41 | \( 1 + 5.71T + 41T^{2} \) |
| 43 | \( 1 - 2.30iT - 43T^{2} \) |
| 47 | \( 1 - 6.88iT - 47T^{2} \) |
| 53 | \( 1 - 6.76iT - 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 - 9.25T + 61T^{2} \) |
| 67 | \( 1 + 15.7iT - 67T^{2} \) |
| 71 | \( 1 + 5.25T + 71T^{2} \) |
| 73 | \( 1 + 6.03iT - 73T^{2} \) |
| 79 | \( 1 - 1.35T + 79T^{2} \) |
| 83 | \( 1 - 8.59iT - 83T^{2} \) |
| 89 | \( 1 - 8.84T + 89T^{2} \) |
| 97 | \( 1 - 8.01iT - 97T^{2} \) |
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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
−10.29599892239293236990260091064, −9.490850213373620302120742384158, −8.841228739558967446460604627488, −7.69635587893771773277357255576, −6.71873195169460356613924500547, −6.27123942616005609087928573102, −4.97913390017172419708668479224, −4.02914924577921089580235413062, −2.85227862607201372036425427034, −1.80494337868896314017923851487,
0.813099031822065403653861895907, 1.89759808271891757399838504772, 3.73366149073663417467170563976, 4.39048881424680461662240242204, 5.41333861686902724138172428447, 6.51733356743388529764650207641, 7.29715948927124687803687852337, 8.239099585407007754368091423777, 8.896535236740795019298026212540, 10.12985566865501574659036750871