| L(s) = 1 | + 0.0546·3-s − 0.338·5-s − 3.60·7-s − 2.99·9-s − 11-s − 5.39·13-s − 0.0184·15-s + 0.824·17-s − 2.90·19-s − 0.196·21-s − 6.59·23-s − 4.88·25-s − 0.327·27-s + 0.0420·29-s − 7.35·31-s − 0.0546·33-s + 1.21·35-s + 8.48·37-s − 0.294·39-s − 7.80·41-s + 6.06·43-s + 1.01·45-s + 7.71·47-s + 5.97·49-s + 0.0450·51-s − 7.32·53-s + 0.338·55-s + ⋯ |
| L(s) = 1 | + 0.0315·3-s − 0.151·5-s − 1.36·7-s − 0.999·9-s − 0.301·11-s − 1.49·13-s − 0.00476·15-s + 0.199·17-s − 0.666·19-s − 0.0429·21-s − 1.37·23-s − 0.977·25-s − 0.0630·27-s + 0.00780·29-s − 1.32·31-s − 0.00950·33-s + 0.205·35-s + 1.39·37-s − 0.0471·39-s − 1.21·41-s + 0.925·43-s + 0.151·45-s + 1.12·47-s + 0.853·49-s + 0.00630·51-s − 1.00·53-s + 0.0455·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3238050197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3238050197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0546T + 3T^{2} \) |
| 5 | \( 1 + 0.338T + 5T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 - 0.824T + 17T^{2} \) |
| 19 | \( 1 + 2.90T + 19T^{2} \) |
| 23 | \( 1 + 6.59T + 23T^{2} \) |
| 29 | \( 1 - 0.0420T + 29T^{2} \) |
| 31 | \( 1 + 7.35T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 + 7.80T + 41T^{2} \) |
| 43 | \( 1 - 6.06T + 43T^{2} \) |
| 47 | \( 1 - 7.71T + 47T^{2} \) |
| 53 | \( 1 + 7.32T + 53T^{2} \) |
| 59 | \( 1 - 9.55T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 7.26T + 67T^{2} \) |
| 71 | \( 1 + 8.18T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + 1.75T + 79T^{2} \) |
| 83 | \( 1 - 7.72T + 83T^{2} \) |
| 89 | \( 1 + 9.60T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.020447981684627901099323305454, −7.39468474337111060210491493135, −6.65472757736248478367953024637, −5.85481916134394020205071442390, −5.45472238621069684228188839295, −4.30366610417003718271245562209, −3.61740984569594859047584054703, −2.70892685300345981027947160696, −2.16078239747302759364618510777, −0.27140880845296395909043727845,
0.27140880845296395909043727845, 2.16078239747302759364618510777, 2.70892685300345981027947160696, 3.61740984569594859047584054703, 4.30366610417003718271245562209, 5.45472238621069684228188839295, 5.85481916134394020205071442390, 6.65472757736248478367953024637, 7.39468474337111060210491493135, 8.020447981684627901099323305454
