L(s) = 1 | + 1.58·2-s − 2.11·3-s + 0.524·4-s − 3.35·6-s − 3.95·7-s − 2.34·8-s + 1.46·9-s + 2.44·11-s − 1.10·12-s + 4.82·13-s − 6.28·14-s − 4.77·16-s − 2.96·17-s + 2.32·18-s + 6.29·19-s + 8.35·21-s + 3.89·22-s − 2.58·23-s + 4.95·24-s + 7.67·26-s + 3.25·27-s − 2.07·28-s + 6.54·29-s − 7.41·31-s − 2.89·32-s − 5.17·33-s − 4.70·34-s + ⋯ |
L(s) = 1 | + 1.12·2-s − 1.21·3-s + 0.262·4-s − 1.36·6-s − 1.49·7-s − 0.828·8-s + 0.486·9-s + 0.738·11-s − 0.319·12-s + 1.33·13-s − 1.67·14-s − 1.19·16-s − 0.717·17-s + 0.546·18-s + 1.44·19-s + 1.82·21-s + 0.829·22-s − 0.538·23-s + 1.01·24-s + 1.50·26-s + 0.625·27-s − 0.392·28-s + 1.21·29-s − 1.33·31-s − 0.511·32-s − 0.900·33-s − 0.806·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 - 1.58T + 2T^{2} \) |
| 3 | \( 1 + 2.11T + 3T^{2} \) |
| 7 | \( 1 + 3.95T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 2.96T + 17T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 + 2.58T + 23T^{2} \) |
| 29 | \( 1 - 6.54T + 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 - 4.28T + 37T^{2} \) |
| 41 | \( 1 + 2.65T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 + 4.42T + 59T^{2} \) |
| 61 | \( 1 - 1.55T + 61T^{2} \) |
| 67 | \( 1 + 5.91T + 67T^{2} \) |
| 71 | \( 1 + 2.90T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 1.75T + 79T^{2} \) |
| 83 | \( 1 - 2.97T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−7.23306691443901724770001221951, −6.59318632228266605334270008530, −6.10700006633934162133082158043, −5.71193672942104510308491744543, −4.93800482418323882833092598971, −4.00697190545191468055184394878, −3.53716265542903861387057828084, −2.72603787543965500363781164054, −1.10193455918127909188301833181, 0,
1.10193455918127909188301833181, 2.72603787543965500363781164054, 3.53716265542903861387057828084, 4.00697190545191468055184394878, 4.93800482418323882833092598971, 5.71193672942104510308491744543, 6.10700006633934162133082158043, 6.59318632228266605334270008530, 7.23306691443901724770001221951