| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 3.44·7-s − 8-s + 9-s + 10-s + 4.24·11-s + 12-s − 3.44·14-s − 15-s + 16-s + 6.78·17-s − 18-s + 6.26·19-s − 20-s + 3.44·21-s − 4.24·22-s − 1.30·23-s − 24-s + 25-s + 27-s + 3.44·28-s − 9.14·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.30·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.28·11-s + 0.288·12-s − 0.920·14-s − 0.258·15-s + 0.250·16-s + 1.64·17-s − 0.235·18-s + 1.43·19-s − 0.223·20-s + 0.751·21-s − 0.905·22-s − 0.272·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.651·28-s − 1.69·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.439027360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.439027360\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 3.44T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 17 | \( 1 - 6.78T + 17T^{2} \) |
| 19 | \( 1 - 6.26T + 19T^{2} \) |
| 23 | \( 1 + 1.30T + 23T^{2} \) |
| 29 | \( 1 + 9.14T + 29T^{2} \) |
| 31 | \( 1 - 3.75T + 31T^{2} \) |
| 37 | \( 1 - 6.82T + 37T^{2} \) |
| 41 | \( 1 - 4.26T + 41T^{2} \) |
| 43 | \( 1 - 3.07T + 43T^{2} \) |
| 47 | \( 1 - 7.76T + 47T^{2} \) |
| 53 | \( 1 + 8.93T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 2.53T + 61T^{2} \) |
| 67 | \( 1 + 0.0760T + 67T^{2} \) |
| 71 | \( 1 - 0.374T + 71T^{2} \) |
| 73 | \( 1 + 16.7T + 73T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 + 0.740T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−8.016097468673954850684336616118, −7.71662317176023335906064658460, −7.26041152112841639190394269275, −6.10926138365097511262815063521, −5.38851262820302847999246857815, −4.37761402083730720886328663899, −3.66095749334016510783173793465, −2.80356100072758271647331762737, −1.55986512162802956482381395759, −1.07412132227436685376947686481,
1.07412132227436685376947686481, 1.55986512162802956482381395759, 2.80356100072758271647331762737, 3.66095749334016510783173793465, 4.37761402083730720886328663899, 5.38851262820302847999246857815, 6.10926138365097511262815063521, 7.26041152112841639190394269275, 7.71662317176023335906064658460, 8.016097468673954850684336616118
