| L(s) = 1 | + (−1.95 − 0.365i)2-s + (−0.294 + 3.17i)3-s + (1.81 + 0.703i)4-s + (1.08 − 1.44i)5-s + (1.73 − 6.09i)6-s + (−4.06 + 2.51i)7-s + (0.0866 + 0.0536i)8-s + (−7.05 − 1.31i)9-s + (−2.65 + 2.41i)10-s + (−0.631 − 0.118i)11-s + (−2.77 + 5.56i)12-s + (−1.14 + 0.707i)13-s + (8.86 − 3.43i)14-s + (4.25 + 3.87i)15-s + (−3.02 − 2.76i)16-s + (0.533 + 1.87i)17-s + ⋯ |
| L(s) = 1 | + (−1.38 − 0.258i)2-s + (−0.169 + 1.83i)3-s + (0.908 + 0.351i)4-s + (0.486 − 0.644i)5-s + (0.708 − 2.48i)6-s + (−1.53 + 0.952i)7-s + (0.0306 + 0.0189i)8-s + (−2.35 − 0.439i)9-s + (−0.838 + 0.764i)10-s + (−0.190 − 0.0356i)11-s + (−0.799 + 1.60i)12-s + (−0.316 + 0.196i)13-s + (2.37 − 0.918i)14-s + (1.09 + 1.00i)15-s + (−0.757 − 0.690i)16-s + (0.129 + 0.454i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.103710 + 0.333103i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.103710 + 0.333103i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (-9.27 - 4.10i)T \) |
| good | 2 | \( 1 + (1.95 + 0.365i)T + (1.86 + 0.722i)T^{2} \) |
| 3 | \( 1 + (0.294 - 3.17i)T + (-2.94 - 0.551i)T^{2} \) |
| 5 | \( 1 + (-1.08 + 1.44i)T + (-1.36 - 4.80i)T^{2} \) |
| 7 | \( 1 + (4.06 - 2.51i)T + (3.12 - 6.26i)T^{2} \) |
| 11 | \( 1 + (0.631 + 0.118i)T + (10.2 + 3.97i)T^{2} \) |
| 13 | \( 1 + (1.14 - 0.707i)T + (5.79 - 11.6i)T^{2} \) |
| 17 | \( 1 + (-0.533 - 1.87i)T + (-14.4 + 8.94i)T^{2} \) |
| 19 | \( 1 + (-0.305 - 3.29i)T + (-18.6 + 3.49i)T^{2} \) |
| 23 | \( 1 + (-4.12 + 0.770i)T + (21.4 - 8.30i)T^{2} \) |
| 29 | \( 1 + (-1.27 + 1.68i)T + (-7.93 - 27.8i)T^{2} \) |
| 31 | \( 1 + (-3.59 - 3.27i)T + (2.86 + 30.8i)T^{2} \) |
| 37 | \( 1 + (0.823 - 1.65i)T + (-22.2 - 29.5i)T^{2} \) |
| 41 | \( 1 + (-5.53 - 7.33i)T + (-11.2 + 39.4i)T^{2} \) |
| 43 | \( 1 + (-4.19 - 8.42i)T + (-25.9 + 34.3i)T^{2} \) |
| 47 | \( 1 + 6.91T + 47T^{2} \) |
| 53 | \( 1 + (0.903 + 9.75i)T + (-52.0 + 9.73i)T^{2} \) |
| 59 | \( 1 + (0.909 + 0.562i)T + (26.2 + 52.8i)T^{2} \) |
| 61 | \( 1 + (-1.46 - 5.15i)T + (-51.8 + 32.1i)T^{2} \) |
| 67 | \( 1 + (4.11 + 2.54i)T + (29.8 + 59.9i)T^{2} \) |
| 71 | \( 1 + (6.21 + 8.23i)T + (-19.4 + 68.2i)T^{2} \) |
| 73 | \( 1 + (-1.38 - 1.82i)T + (-19.9 + 70.2i)T^{2} \) |
| 79 | \( 1 + (-4.76 + 6.31i)T + (-21.6 - 75.9i)T^{2} \) |
| 83 | \( 1 + (6.10 - 3.77i)T + (36.9 - 74.2i)T^{2} \) |
| 89 | \( 1 + (3.29 - 1.27i)T + (65.7 - 59.9i)T^{2} \) |
| 97 | \( 1 + (-0.603 + 2.12i)T + (-82.4 - 51.0i)T^{2} \) |
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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
−14.64052952430752940960757561409, −12.98481884492036968176804711986, −11.69854023136695541482647409471, −10.47707597108999945835946079655, −9.690817988506563052425688487925, −9.310975258805804859948573519892, −8.428484581827866685077915978246, −6.13411525777885741097361115516, −4.89341188222157934175205227939, −3.02463593907285924825301006706,
0.62068502426915281536769557134, 2.67493031943511534703366524210, 6.18277064233471369921860193586, 7.04738657811420933467969282243, 7.42936508642045531729890720339, 8.928661156620121943005746011020, 10.05922068581322220721610951543, 11.01069721377476277992469857544, 12.53060937374748150960400054986, 13.35218152941168235096045404812
