| L(s) = 1 | − 1.74·2-s + 1.05·4-s + 0.963·7-s + 1.64·8-s − 2.88·11-s − 1.65·13-s − 1.68·14-s − 4.99·16-s − 6.95·17-s − 2.39·19-s + 5.04·22-s + 0.0129·23-s + 2.90·26-s + 1.02·28-s + 7.63·29-s + 2.66·31-s + 5.44·32-s + 12.1·34-s + 37-s + 4.18·38-s − 1.52·41-s + 4.22·43-s − 3.05·44-s − 0.0226·46-s + 11.0·47-s − 6.07·49-s − 1.75·52-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.529·4-s + 0.364·7-s + 0.581·8-s − 0.869·11-s − 0.459·13-s − 0.450·14-s − 1.24·16-s − 1.68·17-s − 0.549·19-s + 1.07·22-s + 0.00270·23-s + 0.568·26-s + 0.192·28-s + 1.41·29-s + 0.478·31-s + 0.963·32-s + 2.08·34-s + 0.164·37-s + 0.679·38-s − 0.238·41-s + 0.643·43-s − 0.460·44-s − 0.00334·46-s + 1.61·47-s − 0.867·49-s − 0.243·52-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.74T + 2T^{2} \) |
| 7 | \( 1 - 0.963T + 7T^{2} \) |
| 11 | \( 1 + 2.88T + 11T^{2} \) |
| 13 | \( 1 + 1.65T + 13T^{2} \) |
| 17 | \( 1 + 6.95T + 17T^{2} \) |
| 19 | \( 1 + 2.39T + 19T^{2} \) |
| 23 | \( 1 - 0.0129T + 23T^{2} \) |
| 29 | \( 1 - 7.63T + 29T^{2} \) |
| 31 | \( 1 - 2.66T + 31T^{2} \) |
| 41 | \( 1 + 1.52T + 41T^{2} \) |
| 43 | \( 1 - 4.22T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 9.51T + 53T^{2} \) |
| 59 | \( 1 - 5.18T + 59T^{2} \) |
| 61 | \( 1 + 0.854T + 61T^{2} \) |
| 67 | \( 1 - 9.33T + 67T^{2} \) |
| 71 | \( 1 + 2.93T + 71T^{2} \) |
| 73 | \( 1 + 3.04T + 73T^{2} \) |
| 79 | \( 1 - 8.24T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 8.75T + 89T^{2} \) |
| 97 | \( 1 + 2.99T + 97T^{2} \) |
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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
−7.66513461834745612071260965034, −6.94272140931865803185399204243, −6.39865324286365133488751629984, −5.29205848449379205753464598084, −4.64092352015168269816113686417, −4.05470400641630650804426193759, −2.56897640256488306426556974044, −2.21155143395495969770539875107, −0.983741517676222834149348747762, 0,
0.983741517676222834149348747762, 2.21155143395495969770539875107, 2.56897640256488306426556974044, 4.05470400641630650804426193759, 4.64092352015168269816113686417, 5.29205848449379205753464598084, 6.39865324286365133488751629984, 6.94272140931865803185399204243, 7.66513461834745612071260965034
