| L(s) = 1 | + 2.69·2-s − 3-s + 5.24·4-s − 1.04·5-s − 2.69·6-s + 0.554·7-s + 8.74·8-s + 9-s − 2.82·10-s + 2.91·11-s − 5.24·12-s + 1.49·14-s + 1.04·15-s + 13.0·16-s − 4.85·17-s + 2.69·18-s − 0.753·19-s − 5.50·20-s − 0.554·21-s + 7.83·22-s + 5.76·23-s − 8.74·24-s − 3.89·25-s − 27-s + 2.91·28-s − 1.91·29-s + 2.82·30-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 0.577·3-s + 2.62·4-s − 0.469·5-s − 1.09·6-s + 0.209·7-s + 3.09·8-s + 0.333·9-s − 0.892·10-s + 0.877·11-s − 1.51·12-s + 0.399·14-s + 0.270·15-s + 3.25·16-s − 1.17·17-s + 0.634·18-s − 0.172·19-s − 1.23·20-s − 0.121·21-s + 1.67·22-s + 1.20·23-s − 1.78·24-s − 0.779·25-s − 0.192·27-s + 0.550·28-s − 0.355·29-s + 0.515·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.606997947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.606997947\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 5 | \( 1 + 1.04T + 5T^{2} \) |
| 7 | \( 1 - 0.554T + 7T^{2} \) |
| 11 | \( 1 - 2.91T + 11T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 + 0.753T + 19T^{2} \) |
| 23 | \( 1 - 5.76T + 23T^{2} \) |
| 29 | \( 1 + 1.91T + 29T^{2} \) |
| 31 | \( 1 + 9.51T + 31T^{2} \) |
| 37 | \( 1 - 5.75T + 37T^{2} \) |
| 41 | \( 1 - 4.91T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 0.753T + 47T^{2} \) |
| 53 | \( 1 + 7.58T + 53T^{2} \) |
| 59 | \( 1 - 4.09T + 59T^{2} \) |
| 61 | \( 1 + 3.42T + 61T^{2} \) |
| 67 | \( 1 + 1.87T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 1.33T + 79T^{2} \) |
| 83 | \( 1 - 2.64T + 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−11.39935390333636936449622323814, −10.67509737187974223825622289028, −9.207476267830635728608549577461, −7.69722221732747243271926069083, −6.82303400118216542485672566361, −6.13159882277292972167793745591, −5.06275700736903279327421974501, −4.29783117408011316016719717866, −3.41217922415537127594478474991, −1.87187510316131161114710467104,
1.87187510316131161114710467104, 3.41217922415537127594478474991, 4.29783117408011316016719717866, 5.06275700736903279327421974501, 6.13159882277292972167793745591, 6.82303400118216542485672566361, 7.69722221732747243271926069083, 9.207476267830635728608549577461, 10.67509737187974223825622289028, 11.39935390333636936449622323814
