L(s) = 1 | + 1.41·3-s − 5-s + 1.00·9-s + 1.41·13-s − 1.41·15-s + 19-s + 25-s − 1.41·37-s + 2.00·39-s − 1.00·45-s + 49-s − 1.41·53-s + 1.41·57-s − 1.41·65-s − 1.41·67-s + 1.41·75-s − 0.999·81-s − 95-s − 1.41·97-s − 1.41·103-s − 1.41·107-s − 2.00·111-s + 1.41·113-s + 1.41·117-s + ⋯ |
L(s) = 1 | + 1.41·3-s − 5-s + 1.00·9-s + 1.41·13-s − 1.41·15-s + 19-s + 25-s − 1.41·37-s + 2.00·39-s − 1.00·45-s + 49-s − 1.41·53-s + 1.41·57-s − 1.41·65-s − 1.41·67-s + 1.41·75-s − 0.999·81-s − 95-s − 1.41·97-s − 1.41·103-s − 1.41·107-s − 2.00·111-s + 1.41·113-s + 1.41·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.567611674\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.567611674\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.41T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.41T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−9.333758392986555810566549022775, −8.768366917604205082708066046311, −8.142098538207784704121976491787, −7.53483326984986013544304140303, −6.69205321015119439143585820856, −5.48376300487681058471845699583, −4.25691348345654187211046544339, −3.53075132896040688641337643238, −2.92013503580586984101221815322, −1.47050220859465655908048266546,
1.47050220859465655908048266546, 2.92013503580586984101221815322, 3.53075132896040688641337643238, 4.25691348345654187211046544339, 5.48376300487681058471845699583, 6.69205321015119439143585820856, 7.53483326984986013544304140303, 8.142098538207784704121976491787, 8.768366917604205082708066046311, 9.333758392986555810566549022775