| L(s) = 1 | − 1.90·2-s − 0.688·3-s + 1.62·4-s + 1.31·6-s + 7-s + 0.719·8-s − 2.52·9-s − 0.903·11-s − 1.11·12-s − 0.622·13-s − 1.90·14-s − 4.61·16-s − 1.31·17-s + 4.80·18-s + 7.05·19-s − 0.688·21-s + 1.71·22-s − 23-s − 0.495·24-s + 1.18·26-s + 3.80·27-s + 1.62·28-s + 1.52·29-s − 3.54·31-s + 7.34·32-s + 0.622·33-s + 2.49·34-s + ⋯ |
| L(s) = 1 | − 1.34·2-s − 0.397·3-s + 0.811·4-s + 0.535·6-s + 0.377·7-s + 0.254·8-s − 0.841·9-s − 0.272·11-s − 0.322·12-s − 0.172·13-s − 0.508·14-s − 1.15·16-s − 0.317·17-s + 1.13·18-s + 1.61·19-s − 0.150·21-s + 0.366·22-s − 0.208·23-s − 0.101·24-s + 0.232·26-s + 0.732·27-s + 0.306·28-s + 0.283·29-s − 0.636·31-s + 1.29·32-s + 0.108·33-s + 0.427·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5572568549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5572568549\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.90T + 2T^{2} \) |
| 3 | \( 1 + 0.688T + 3T^{2} \) |
| 11 | \( 1 + 0.903T + 11T^{2} \) |
| 13 | \( 1 + 0.622T + 13T^{2} \) |
| 17 | \( 1 + 1.31T + 17T^{2} \) |
| 19 | \( 1 - 7.05T + 19T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 + 3.54T + 31T^{2} \) |
| 37 | \( 1 + 2.42T + 37T^{2} \) |
| 41 | \( 1 + 4.75T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + 9.73T + 47T^{2} \) |
| 53 | \( 1 - 6.85T + 53T^{2} \) |
| 59 | \( 1 - 9.97T + 59T^{2} \) |
| 61 | \( 1 - 7.87T + 61T^{2} \) |
| 67 | \( 1 - 5.76T + 67T^{2} \) |
| 71 | \( 1 + 7.61T + 71T^{2} \) |
| 73 | \( 1 - 1.86T + 73T^{2} \) |
| 79 | \( 1 - 7.65T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 2.98T + 89T^{2} \) |
| 97 | \( 1 - 2.49T + 97T^{2} \) |
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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
−8.425308183133298741127199859433, −7.956646014302293889850674402183, −7.14838417074017457430152474628, −6.51887004167577004271702838700, −5.33930079429844532811989389245, −5.03299347904650089768613060712, −3.72724678193834488615221403709, −2.66202331167526240277853732272, −1.63738074433197946410813916656, −0.54471909104086818357053693887,
0.54471909104086818357053693887, 1.63738074433197946410813916656, 2.66202331167526240277853732272, 3.72724678193834488615221403709, 5.03299347904650089768613060712, 5.33930079429844532811989389245, 6.51887004167577004271702838700, 7.14838417074017457430152474628, 7.956646014302293889850674402183, 8.425308183133298741127199859433
