| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 4.60·7-s − 8-s + 9-s + 10-s − 11-s + 12-s − 6.60·13-s − 4.60·14-s − 15-s + 16-s + 17-s − 18-s − 6.60·19-s − 20-s + 4.60·21-s + 22-s + 2.60·23-s − 24-s + 25-s + 6.60·26-s + 27-s + 4.60·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.74·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s − 1.83·13-s − 1.23·14-s − 0.258·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s − 1.51·19-s − 0.223·20-s + 1.00·21-s + 0.213·22-s + 0.543·23-s − 0.204·24-s + 0.200·25-s + 1.29·26-s + 0.192·27-s + 0.870·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4.60T + 7T^{2} \) |
| 13 | \( 1 + 6.60T + 13T^{2} \) |
| 19 | \( 1 + 6.60T + 19T^{2} \) |
| 23 | \( 1 - 2.60T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 9.21T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 + 5.21T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 8.60T + 53T^{2} \) |
| 59 | \( 1 + 3.39T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 8.60T + 71T^{2} \) |
| 73 | \( 1 + 1.39T + 73T^{2} \) |
| 79 | \( 1 + 6.60T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 1.21T + 97T^{2} \) |
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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
−7.86354272231428348680156130691, −7.39028888963835754331503056706, −6.73573753377005523858574591547, −5.48698972548728584931224410300, −4.75601841967884716540506891250, −4.25108878134129653544110239356, −2.94906274991163531189299227834, −2.21667261034562502990386025769, −1.46567973326919740923751247913, 0,
1.46567973326919740923751247913, 2.21667261034562502990386025769, 2.94906274991163531189299227834, 4.25108878134129653544110239356, 4.75601841967884716540506891250, 5.48698972548728584931224410300, 6.73573753377005523858574591547, 7.39028888963835754331503056706, 7.86354272231428348680156130691
