| L(s) = 1 | + (1.49 − 1.66i)5-s − 4.78·7-s + (1.58 + 1.14i)11-s + (−0.873 + 0.634i)13-s + (1.17 − 3.61i)17-s + (1.31 − 4.04i)19-s + (−4.74 − 3.44i)23-s + (−0.522 − 4.97i)25-s + (−3.26 − 10.0i)29-s + (−1.33 + 4.10i)31-s + (−7.15 + 7.94i)35-s + (−4.57 + 3.32i)37-s + (0.694 − 0.504i)41-s − 10.8·43-s + (−0.927 − 2.85i)47-s + ⋯ |
| L(s) = 1 | + (0.669 − 0.743i)5-s − 1.80·7-s + (0.477 + 0.346i)11-s + (−0.242 + 0.176i)13-s + (0.285 − 0.877i)17-s + (0.301 − 0.927i)19-s + (−0.989 − 0.718i)23-s + (−0.104 − 0.994i)25-s + (−0.605 − 1.86i)29-s + (−0.239 + 0.737i)31-s + (−1.20 + 1.34i)35-s + (−0.752 + 0.546i)37-s + (0.108 − 0.0788i)41-s − 1.65·43-s + (−0.135 − 0.416i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.396304 - 0.798350i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.396304 - 0.798350i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.49 + 1.66i)T \) |
| good | 7 | \( 1 + 4.78T + 7T^{2} \) |
| 11 | \( 1 + (-1.58 - 1.14i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.873 - 0.634i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.17 + 3.61i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.31 + 4.04i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (4.74 + 3.44i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (3.26 + 10.0i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.33 - 4.10i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (4.57 - 3.32i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.694 + 0.504i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + (0.927 + 2.85i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.30 + 4.01i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.85 - 2.80i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (2.93 + 2.13i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.14 + 6.59i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.70 - 11.4i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-13.7 - 9.96i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.04 + 6.29i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.797 + 2.45i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (0.673 + 0.489i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.81 + 8.67i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.673010170252311806951974844236, −9.301578867211627064057062886604, −8.290917978135032383022010576443, −6.97898977977135846450866102966, −6.44537456220484486556181568471, −5.49813121414092445850620412476, −4.46783854682799277461747572402, −3.30985239778324370172691487678, −2.17614025498434116938283831837, −0.40002406496900186067314329014,
1.80874729794306937267593840307, 3.29990854050427027932397915327, 3.61993620982835888827964359347, 5.51957863748972840728183648195, 6.14193386449421524950987012265, 6.80346661483059680222614992052, 7.73757419271364039999085383440, 9.007178847420638264639693267629, 9.698849157988490859808234319360, 10.22159675916632831236769690335
