| L(s) = 1 | − 1.67·3-s + 3.28·5-s − 7-s − 0.193·9-s + 2.15·11-s + 2.96·13-s − 5.50·15-s − 0.637·17-s + 1.67·21-s + 23-s + 5.80·25-s + 5.35·27-s + 3.19·29-s − 5.59·31-s − 3.61·33-s − 3.28·35-s + 10.9·37-s − 4.96·39-s + 1.22·41-s − 5.73·43-s − 0.637·45-s − 4.63·47-s + 49-s + 1.06·51-s − 5.22·53-s + 7.08·55-s + 1.98·59-s + ⋯ |
| L(s) = 1 | − 0.967·3-s + 1.47·5-s − 0.377·7-s − 0.0646·9-s + 0.650·11-s + 0.821·13-s − 1.42·15-s − 0.154·17-s + 0.365·21-s + 0.208·23-s + 1.16·25-s + 1.02·27-s + 0.593·29-s − 1.00·31-s − 0.628·33-s − 0.555·35-s + 1.80·37-s − 0.794·39-s + 0.191·41-s − 0.875·43-s − 0.0950·45-s − 0.676·47-s + 0.142·49-s + 0.149·51-s − 0.717·53-s + 0.955·55-s + 0.258·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.534571718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.534571718\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 1.67T + 3T^{2} \) |
| 5 | \( 1 - 3.28T + 5T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 + 0.637T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 - 3.19T + 29T^{2} \) |
| 31 | \( 1 + 5.59T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 1.22T + 41T^{2} \) |
| 43 | \( 1 + 5.73T + 43T^{2} \) |
| 47 | \( 1 + 4.63T + 47T^{2} \) |
| 53 | \( 1 + 5.22T + 53T^{2} \) |
| 59 | \( 1 - 1.98T + 59T^{2} \) |
| 61 | \( 1 - 5.80T + 61T^{2} \) |
| 67 | \( 1 - 8.08T + 67T^{2} \) |
| 71 | \( 1 - 6.70T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 7.76T + 79T^{2} \) |
| 83 | \( 1 - 0.962T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−9.633357811356943364419293075499, −9.131413902780077925068636753444, −8.132151362952353897726325371997, −6.66313753859540812405418132620, −6.34499045003953149734510857282, −5.63249663012454666436272289381, −4.84838783136515717940890214095, −3.52171008825010458491114133934, −2.24646305852503286992868834407, −1.00076015132035651466054313744,
1.00076015132035651466054313744, 2.24646305852503286992868834407, 3.52171008825010458491114133934, 4.84838783136515717940890214095, 5.63249663012454666436272289381, 6.34499045003953149734510857282, 6.66313753859540812405418132620, 8.132151362952353897726325371997, 9.131413902780077925068636753444, 9.633357811356943364419293075499
