| L(s) = 1 | − 1.77·3-s + 0.595·5-s − 7-s + 0.158·9-s − 6.33·11-s − 1.33·13-s − 1.05·15-s + 6.14·17-s + 19-s + 1.77·21-s + 0.680·23-s − 4.64·25-s + 5.05·27-s − 0.562·29-s − 7.08·31-s + 11.2·33-s − 0.595·35-s − 3.86·37-s + 2.37·39-s + 5.17·41-s − 6.54·43-s + 0.0942·45-s + 5.37·47-s + 49-s − 10.9·51-s − 11.0·53-s − 3.77·55-s + ⋯ |
| L(s) = 1 | − 1.02·3-s + 0.266·5-s − 0.377·7-s + 0.0527·9-s − 1.91·11-s − 0.370·13-s − 0.273·15-s + 1.49·17-s + 0.229·19-s + 0.387·21-s + 0.141·23-s − 0.929·25-s + 0.971·27-s − 0.104·29-s − 1.27·31-s + 1.96·33-s − 0.100·35-s − 0.635·37-s + 0.380·39-s + 0.808·41-s − 0.998·43-s + 0.0140·45-s + 0.784·47-s + 0.142·49-s − 1.52·51-s − 1.51·53-s − 0.508·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5105778757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5105778757\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.77T + 3T^{2} \) |
| 5 | \( 1 - 0.595T + 5T^{2} \) |
| 11 | \( 1 + 6.33T + 11T^{2} \) |
| 13 | \( 1 + 1.33T + 13T^{2} \) |
| 17 | \( 1 - 6.14T + 17T^{2} \) |
| 23 | \( 1 - 0.680T + 23T^{2} \) |
| 29 | \( 1 + 0.562T + 29T^{2} \) |
| 31 | \( 1 + 7.08T + 31T^{2} \) |
| 37 | \( 1 + 3.86T + 37T^{2} \) |
| 41 | \( 1 - 5.17T + 41T^{2} \) |
| 43 | \( 1 + 6.54T + 43T^{2} \) |
| 47 | \( 1 - 5.37T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 6.32T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 4.12T + 73T^{2} \) |
| 79 | \( 1 + 6.21T + 79T^{2} \) |
| 83 | \( 1 + 3.14T + 83T^{2} \) |
| 89 | \( 1 + 1.33T + 89T^{2} \) |
| 97 | \( 1 + 2.06T + 97T^{2} \) |
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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
−7.60956913971782393445019301174, −7.20918929678146772436883677353, −6.14904247331892997332156347832, −5.53886069923595298572524755070, −5.37475638908140115226249238914, −4.49885456305601202983006206858, −3.31991884101084332601411198935, −2.76064973780287042290941075583, −1.66313313364508983070485539940, −0.35842584368134353738288913281,
0.35842584368134353738288913281, 1.66313313364508983070485539940, 2.76064973780287042290941075583, 3.31991884101084332601411198935, 4.49885456305601202983006206858, 5.37475638908140115226249238914, 5.53886069923595298572524755070, 6.14904247331892997332156347832, 7.20918929678146772436883677353, 7.60956913971782393445019301174
