| L(s) = 1 | + 2-s + 3-s + 4-s + 3.79·5-s + 6-s + 8-s + 9-s + 3.79·10-s + 11-s + 12-s + 0.361·13-s + 3.79·15-s + 16-s − 4.11·17-s + 18-s + 4.15·19-s + 3.79·20-s + 22-s + 0.542·23-s + 24-s + 9.42·25-s + 0.361·26-s + 27-s + 0.767·29-s + 3.79·30-s − 8.80·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.69·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.20·10-s + 0.301·11-s + 0.288·12-s + 0.100·13-s + 0.980·15-s + 0.250·16-s − 0.998·17-s + 0.235·18-s + 0.954·19-s + 0.849·20-s + 0.213·22-s + 0.113·23-s + 0.204·24-s + 1.88·25-s + 0.0707·26-s + 0.192·27-s + 0.142·29-s + 0.693·30-s − 1.58·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.214489135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.214489135\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 3.79T + 5T^{2} \) |
| 13 | \( 1 - 0.361T + 13T^{2} \) |
| 17 | \( 1 + 4.11T + 17T^{2} \) |
| 19 | \( 1 - 4.15T + 19T^{2} \) |
| 23 | \( 1 - 0.542T + 23T^{2} \) |
| 29 | \( 1 - 0.767T + 29T^{2} \) |
| 31 | \( 1 + 8.80T + 31T^{2} \) |
| 37 | \( 1 - 2.28T + 37T^{2} \) |
| 41 | \( 1 - 9.13T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 2.98T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 - 2.25T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 0.603T + 67T^{2} \) |
| 71 | \( 1 + 6.87T + 71T^{2} \) |
| 73 | \( 1 - 9.45T + 73T^{2} \) |
| 79 | \( 1 - 0.0321T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 1.29T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−8.831771597696532430106473516609, −7.85266007325249459505794811542, −6.88228872325713884724352083632, −6.39791929540669862703473869905, −5.53318082853644444514427321141, −4.96088753358193560835931489606, −3.91426030742745243696024291662, −2.95915015297857717185863544402, −2.16442400486375776669303336654, −1.40580224513349436952036171564,
1.40580224513349436952036171564, 2.16442400486375776669303336654, 2.95915015297857717185863544402, 3.91426030742745243696024291662, 4.96088753358193560835931489606, 5.53318082853644444514427321141, 6.39791929540669862703473869905, 6.88228872325713884724352083632, 7.85266007325249459505794811542, 8.831771597696532430106473516609
