| L(s) = 1 | − i·3-s + (2.08 − 0.805i)5-s + 3.70i·7-s − 9-s + 3.31·11-s + i·13-s + (−0.805 − 2.08i)15-s + 4.36i·17-s + 5.21·19-s + 3.70·21-s − 4.92i·23-s + (3.70 − 3.36i)25-s + i·27-s − 7.78·29-s − 0.0981·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.932 − 0.360i)5-s + 1.39i·7-s − 0.333·9-s + 0.998·11-s + 0.277i·13-s + (−0.208 − 0.538i)15-s + 1.05i·17-s + 1.19·19-s + 0.807·21-s − 1.02i·23-s + (0.740 − 0.672i)25-s + 0.192i·27-s − 1.44·29-s − 0.0176·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.365680501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.365680501\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.08 + 0.805i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 3.70iT - 7T^{2} \) |
| 11 | \( 1 - 3.31T + 11T^{2} \) |
| 17 | \( 1 - 4.36iT - 17T^{2} \) |
| 19 | \( 1 - 5.21T + 19T^{2} \) |
| 23 | \( 1 + 4.92iT - 23T^{2} \) |
| 29 | \( 1 + 7.78T + 29T^{2} \) |
| 31 | \( 1 + 0.0981T + 31T^{2} \) |
| 37 | \( 1 - 2.92iT - 37T^{2} \) |
| 41 | \( 1 + 0.749T + 41T^{2} \) |
| 43 | \( 1 - 3.78iT - 43T^{2} \) |
| 47 | \( 1 - 5.67iT - 47T^{2} \) |
| 53 | \( 1 + 2.19iT - 53T^{2} \) |
| 59 | \( 1 - 0.108T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 12.4iT - 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 - 9.56iT - 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 16.7iT - 83T^{2} \) |
| 89 | \( 1 - 3.59T + 89T^{2} \) |
| 97 | \( 1 - 4.10iT - 97T^{2} \) |
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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
−8.709449433361620292342844867688, −8.240695888574173144452972209901, −7.08182825765185905023131579900, −6.33525906583198661410644394612, −5.78921593999875608568659333950, −5.18005247093501870769736388676, −4.03520996558239214019560905974, −2.84771743135232027145354908750, −2.01195636560396297365237826057, −1.23680581069162322501315425016,
0.810711765276717075172126639625, 1.93300566586562200950976494017, 3.32466395157566197252823154910, 3.72179159200953132990555009560, 4.86172264056813996188297998890, 5.51177084557634288928614315474, 6.39567490151033727518314959994, 7.28942760637427462152361240401, 7.57176606238020220019373145886, 9.065807137930113471522199446550
