| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 2.44·7-s + 8-s + 9-s − 4.89·11-s − 12-s − 6·13-s + 2.44·14-s + 16-s − 17-s + 18-s + 6.89·19-s − 2.44·21-s − 4.89·22-s − 6.44·23-s − 24-s − 6·26-s − 27-s + 2.44·28-s − 9.34·29-s − 6.44·31-s + 32-s + 4.89·33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.925·7-s + 0.353·8-s + 0.333·9-s − 1.47·11-s − 0.288·12-s − 1.66·13-s + 0.654·14-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 1.58·19-s − 0.534·21-s − 1.04·22-s − 1.34·23-s − 0.204·24-s − 1.17·26-s − 0.192·27-s + 0.462·28-s − 1.73·29-s − 1.15·31-s + 0.176·32-s + 0.852·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 19 | \( 1 - 6.89T + 19T^{2} \) |
| 23 | \( 1 + 6.44T + 23T^{2} \) |
| 29 | \( 1 + 9.34T + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 37 | \( 1 - 0.449T + 37T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 - 2.89T + 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 - 5.79T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 2.44T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 1.55T + 79T^{2} \) |
| 83 | \( 1 + 2.89T + 83T^{2} \) |
| 89 | \( 1 + 1.79T + 89T^{2} \) |
| 97 | \( 1 - 3.79T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.206315397634574447902406415734, −7.45397257341482684004048396476, −7.24214182222067515602782235079, −5.78841642242224113593090383967, −5.30243407290834490264870357348, −4.83414082823285989631889017844, −3.80873989216357019006401529082, −2.60833362751373309240544519149, −1.79684396230397726346931956942, 0,
1.79684396230397726346931956942, 2.60833362751373309240544519149, 3.80873989216357019006401529082, 4.83414082823285989631889017844, 5.30243407290834490264870357348, 5.78841642242224113593090383967, 7.24214182222067515602782235079, 7.45397257341482684004048396476, 8.206315397634574447902406415734
