L(s) = 1 | + 1.31·2-s + 3-s − 0.274·4-s + 1.31·6-s + 4.19·7-s − 2.98·8-s + 9-s − 0.167·11-s − 0.274·12-s − 3.39·13-s + 5.50·14-s − 3.37·16-s + 4.57·17-s + 1.31·18-s + 3.78·19-s + 4.19·21-s − 0.220·22-s + 8.31·23-s − 2.98·24-s − 4.45·26-s + 27-s − 1.15·28-s + 2.74·29-s − 7.71·31-s + 1.54·32-s − 0.167·33-s + 6.00·34-s + ⋯ |
L(s) = 1 | + 0.928·2-s + 0.577·3-s − 0.137·4-s + 0.536·6-s + 1.58·7-s − 1.05·8-s + 0.333·9-s − 0.0505·11-s − 0.0792·12-s − 0.941·13-s + 1.47·14-s − 0.843·16-s + 1.10·17-s + 0.309·18-s + 0.867·19-s + 0.914·21-s − 0.0469·22-s + 1.73·23-s − 0.609·24-s − 0.874·26-s + 0.192·27-s − 0.217·28-s + 0.509·29-s − 1.38·31-s + 0.272·32-s − 0.0292·33-s + 1.03·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.592054772\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.592054772\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 1.31T + 2T^{2} \) |
| 7 | \( 1 - 4.19T + 7T^{2} \) |
| 11 | \( 1 + 0.167T + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 17 | \( 1 - 4.57T + 17T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 23 | \( 1 - 8.31T + 23T^{2} \) |
| 29 | \( 1 - 2.74T + 29T^{2} \) |
| 31 | \( 1 + 7.71T + 31T^{2} \) |
| 37 | \( 1 - 2.07T + 37T^{2} \) |
| 41 | \( 1 + 1.28T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 9.99T + 47T^{2} \) |
| 53 | \( 1 + 1.07T + 53T^{2} \) |
| 59 | \( 1 + 4.95T + 59T^{2} \) |
| 61 | \( 1 - 2.36T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 8.67T + 73T^{2} \) |
| 79 | \( 1 + 2.64T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 4.06T + 89T^{2} \) |
| 97 | \( 1 + 2.78T + 97T^{2} \) |
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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
−9.221738487804797920717557602432, −8.365083614769039401305153106934, −7.68059376986608340756711469214, −6.98007386845473149095295240974, −5.53487024852513171543505448509, −5.11705254067890107105669451845, −4.41851519283913319544985361436, −3.39939060259212624818149311914, −2.54841034615921783530298958530, −1.21537359839890350196407952427,
1.21537359839890350196407952427, 2.54841034615921783530298958530, 3.39939060259212624818149311914, 4.41851519283913319544985361436, 5.11705254067890107105669451845, 5.53487024852513171543505448509, 6.98007386845473149095295240974, 7.68059376986608340756711469214, 8.365083614769039401305153106934, 9.221738487804797920717557602432