| L(s) = 1 | + 0.689·2-s − 2.51·3-s − 1.52·4-s − 1.73·6-s − 7-s − 2.43·8-s + 3.31·9-s + 3.58·11-s + 3.82·12-s − 0.891·13-s − 0.689·14-s + 1.37·16-s − 4.87·17-s + 2.28·18-s − 3.14·19-s + 2.51·21-s + 2.47·22-s − 23-s + 6.10·24-s − 0.614·26-s − 0.779·27-s + 1.52·28-s + 8.91·29-s − 0.999·31-s + 5.80·32-s − 8.99·33-s − 3.36·34-s + ⋯ |
| L(s) = 1 | + 0.487·2-s − 1.45·3-s − 0.761·4-s − 0.707·6-s − 0.377·7-s − 0.859·8-s + 1.10·9-s + 1.07·11-s + 1.10·12-s − 0.247·13-s − 0.184·14-s + 0.342·16-s − 1.18·17-s + 0.538·18-s − 0.720·19-s + 0.548·21-s + 0.526·22-s − 0.208·23-s + 1.24·24-s − 0.120·26-s − 0.149·27-s + 0.288·28-s + 1.65·29-s − 0.179·31-s + 1.02·32-s − 1.56·33-s − 0.576·34-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.689T + 2T^{2} \) |
| 3 | \( 1 + 2.51T + 3T^{2} \) |
| 11 | \( 1 - 3.58T + 11T^{2} \) |
| 13 | \( 1 + 0.891T + 13T^{2} \) |
| 17 | \( 1 + 4.87T + 17T^{2} \) |
| 19 | \( 1 + 3.14T + 19T^{2} \) |
| 29 | \( 1 - 8.91T + 29T^{2} \) |
| 31 | \( 1 + 0.999T + 31T^{2} \) |
| 37 | \( 1 - 5.32T + 37T^{2} \) |
| 41 | \( 1 - 2.66T + 41T^{2} \) |
| 43 | \( 1 + 3.76T + 43T^{2} \) |
| 47 | \( 1 - 6.09T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 + 4.77T + 59T^{2} \) |
| 61 | \( 1 - 3.06T + 61T^{2} \) |
| 67 | \( 1 - 9.01T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 1.73T + 73T^{2} \) |
| 79 | \( 1 - 3.47T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 + 1.02T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.163382491926613583761829714855, −6.86528065331008177979703853648, −6.43272426563125472097673707474, −5.90284076804750236030430602273, −4.98741655940148242328648422991, −4.43653385745926148985436052806, −3.82301251036661399336141353141, −2.56350602008399536902814516509, −1.03076100249017821423717336654, 0,
1.03076100249017821423717336654, 2.56350602008399536902814516509, 3.82301251036661399336141353141, 4.43653385745926148985436052806, 4.98741655940148242328648422991, 5.90284076804750236030430602273, 6.43272426563125472097673707474, 6.86528065331008177979703853648, 8.163382491926613583761829714855
