L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 11-s + 12-s − 4·13-s + 16-s + 3·17-s − 18-s − 19-s − 22-s − 3·23-s − 24-s − 5·25-s + 4·26-s + 27-s − 9·29-s + 2·31-s − 32-s + 33-s − 3·34-s + 36-s − 7·37-s + 38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.229·19-s − 0.213·22-s − 0.625·23-s − 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s − 1.67·29-s + 0.359·31-s − 0.176·32-s + 0.174·33-s − 0.514·34-s + 1/6·36-s − 1.15·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.305720941187786996297649971502, −7.54428724481666134410033113102, −7.17721884504773587226772690413, −6.10424681555187562810349855261, −5.34202267030068632831586779804, −4.21767793617911823901833739858, −3.39441202781466408962518852045, −2.37196678602526111666226831883, −1.57402800329540727348342528650, 0,
1.57402800329540727348342528650, 2.37196678602526111666226831883, 3.39441202781466408962518852045, 4.21767793617911823901833739858, 5.34202267030068632831586779804, 6.10424681555187562810349855261, 7.17721884504773587226772690413, 7.54428724481666134410033113102, 8.305720941187786996297649971502