| L(s) = 1 | + 2-s + i·3-s + 4-s + i·6-s − 4.60·7-s + 8-s − 9-s + i·12-s − 3.60i·13-s − 4.60·14-s + 16-s + 4.60i·17-s − 18-s − 4.60i·19-s − 4.60i·21-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.5·4-s + 0.408i·6-s − 1.74·7-s + 0.353·8-s − 0.333·9-s + 0.288i·12-s − 0.999i·13-s − 1.23·14-s + 0.250·16-s + 1.11i·17-s − 0.235·18-s − 1.05i·19-s − 1.00i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7275423752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7275423752\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 3.60iT \) |
| good | 7 | \( 1 + 4.60T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 4.60iT - 17T^{2} \) |
| 19 | \( 1 + 4.60iT - 19T^{2} \) |
| 23 | \( 1 - 1.39iT - 23T^{2} \) |
| 29 | \( 1 + 4.60T + 29T^{2} \) |
| 31 | \( 1 + 6iT - 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 + 3.21iT - 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 9.21iT - 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 3.21T + 67T^{2} \) |
| 71 | \( 1 + 9.21iT - 71T^{2} \) |
| 73 | \( 1 - 1.39T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 - 15.2iT - 89T^{2} \) |
| 97 | \( 1 - 1.39T + 97T^{2} \) |
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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
−9.133160596343358663695488347411, −8.145304537684110307765091856218, −7.13609483125820731095372293368, −6.37045252614063699244006240248, −5.72112294178280398128632622560, −4.90625525793953494770958442574, −3.56807533425003467440536102940, −3.47469200786579509389468025975, −2.24043676272295136540204710656, −0.19012181532469475818470822057,
1.58760189885370575688140573628, 2.85211539484808619584146735228, 3.44860185356812026048931930270, 4.50999007290869629836975895257, 5.59824211713860969595472237013, 6.35927774018892325852651043252, 6.87609889014198925526995957769, 7.54473255362990282546120780437, 8.727809578819048066326285069188, 9.472836873445225619334620277783
