| L(s) = 1 | + 0.414·2-s − 1.82·4-s + 2·7-s − 1.58·8-s + 2.82·11-s + 0.828·13-s + 0.828·14-s + 3·16-s + 7.65·17-s + 7.24·19-s + 1.17·22-s − 1.58·23-s + 0.343·26-s − 3.65·28-s − 0.828·29-s + 6·31-s + 4.41·32-s + 3.17·34-s + 37-s + 2.99·38-s + 10.6·41-s + 7.24·43-s − 5.17·44-s − 0.656·46-s − 3.17·47-s − 3·49-s − 1.51·52-s + ⋯ |
| L(s) = 1 | + 0.292·2-s − 0.914·4-s + 0.755·7-s − 0.560·8-s + 0.852·11-s + 0.229·13-s + 0.221·14-s + 0.750·16-s + 1.85·17-s + 1.66·19-s + 0.249·22-s − 0.330·23-s + 0.0672·26-s − 0.691·28-s − 0.153·29-s + 1.07·31-s + 0.780·32-s + 0.543·34-s + 0.164·37-s + 0.486·38-s + 1.66·41-s + 1.10·43-s − 0.779·44-s − 0.0968·46-s − 0.462·47-s − 0.428·49-s − 0.210·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.665448432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.665448432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 - 7.24T + 19T^{2} \) |
| 23 | \( 1 + 1.58T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 7.24T + 43T^{2} \) |
| 47 | \( 1 + 3.17T + 47T^{2} \) |
| 53 | \( 1 + 8.65T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 - 3.65T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 5.48T + 73T^{2} \) |
| 79 | \( 1 + 2.07T + 79T^{2} \) |
| 83 | \( 1 + 9.65T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 8.48T + 97T^{2} \) |
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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
−7.79927869541503912284701764905, −7.37008109888527236471030330612, −6.00056838938248724940567854879, −5.84632585547032318732850895232, −4.85339431726469765731056247577, −4.42916190919030827833310243497, −3.47953699507047527364024333615, −3.00031791957490537302322847286, −1.45924967519761447017585929781, −0.892225464574838377402915387179,
0.892225464574838377402915387179, 1.45924967519761447017585929781, 3.00031791957490537302322847286, 3.47953699507047527364024333615, 4.42916190919030827833310243497, 4.85339431726469765731056247577, 5.84632585547032318732850895232, 6.00056838938248724940567854879, 7.37008109888527236471030330612, 7.79927869541503912284701764905
