| L(s) = 1 | + 48i·3-s − 532i·7-s + 1.73e4·9-s + 3.31e4·11-s − 9.96e4i·13-s + 4.43e5i·17-s − 3.57e5·19-s + 2.55e4·21-s + 1.42e5i·23-s + 1.77e6i·27-s − 1.52e6·29-s − 7.32e6·31-s + 1.59e6i·33-s + 2.66e6i·37-s + 4.78e6·39-s + ⋯ |
| L(s) = 1 | + 0.342i·3-s − 0.0837i·7-s + 0.882·9-s + 0.683·11-s − 0.967i·13-s + 1.28i·17-s − 0.628·19-s + 0.0286·21-s + 0.106i·23-s + 0.644i·27-s − 0.401·29-s − 1.42·31-s + 0.233i·33-s + 0.233i·37-s + 0.331·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.649146566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.649146566\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 48iT - 1.96e4T^{2} \) |
| 7 | \( 1 + 532iT - 4.03e7T^{2} \) |
| 11 | \( 1 - 3.31e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.96e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 4.43e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 + 3.57e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.42e5iT - 1.80e12T^{2} \) |
| 29 | \( 1 + 1.52e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.32e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.66e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 7.93e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.11e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 - 1.60e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + 8.78e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 - 1.20e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 9.35e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.93e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 + 4.17e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.50e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 + 9.14e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.52e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 - 1.70e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.09e7iT - 7.60e17T^{2} \) |
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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
−10.17179199952726197123152095172, −9.198377362072425016549738974349, −8.271819195537116423537464040113, −7.29548551000225492541758966831, −6.32315180420718378009043601260, −5.31545785966483708593213917710, −4.14927331687809491590614073356, −3.50739222238263473834701725950, −2.00645149269065615728914807333, −1.05053865508705531363743132617,
0.30753752312250959113727371606, 1.46390409439024678747170708247, 2.30484806609126470055632382287, 3.75723443451676904384276245104, 4.56064948720298721894691367476, 5.76850481623687806240656664989, 6.96663304110574707365977859620, 7.26930288668559868452025476982, 8.718612400831154492635101797633, 9.359709438653822245727434675840
