| L(s) = 1 | − 2.61·2-s − 0.259·3-s + 4.81·4-s − 4.09·5-s + 0.678·6-s + 2.08·7-s − 7.34·8-s − 2.93·9-s + 10.6·10-s − 4.54·11-s − 1.25·12-s − 2.98·13-s − 5.45·14-s + 1.06·15-s + 9.55·16-s + 4.39·17-s + 7.65·18-s − 0.751·19-s − 19.7·20-s − 0.542·21-s + 11.8·22-s − 0.781·23-s + 1.90·24-s + 11.7·25-s + 7.78·26-s + 1.54·27-s + 10.0·28-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 0.149·3-s + 2.40·4-s − 1.83·5-s + 0.276·6-s + 0.789·7-s − 2.59·8-s − 0.977·9-s + 3.37·10-s − 1.37·11-s − 0.361·12-s − 0.826·13-s − 1.45·14-s + 0.274·15-s + 2.38·16-s + 1.06·17-s + 1.80·18-s − 0.172·19-s − 4.40·20-s − 0.118·21-s + 2.52·22-s − 0.162·23-s + 0.389·24-s + 2.35·25-s + 1.52·26-s + 0.296·27-s + 1.89·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02317314173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02317314173\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4001 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + 0.259T + 3T^{2} \) |
| 5 | \( 1 + 4.09T + 5T^{2} \) |
| 7 | \( 1 - 2.08T + 7T^{2} \) |
| 11 | \( 1 + 4.54T + 11T^{2} \) |
| 13 | \( 1 + 2.98T + 13T^{2} \) |
| 17 | \( 1 - 4.39T + 17T^{2} \) |
| 19 | \( 1 + 0.751T + 19T^{2} \) |
| 23 | \( 1 + 0.781T + 23T^{2} \) |
| 29 | \( 1 + 4.78T + 29T^{2} \) |
| 31 | \( 1 + 3.90T + 31T^{2} \) |
| 37 | \( 1 + 6.77T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 0.552T + 43T^{2} \) |
| 47 | \( 1 + 1.35T + 47T^{2} \) |
| 53 | \( 1 + 8.71T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 0.470T + 61T^{2} \) |
| 67 | \( 1 + 8.34T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 7.49T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−8.268326236063130669308822971407, −7.85171910890825771439941783451, −7.53998433678162739301622800888, −6.75292872494400020860570808565, −5.49802359397779796042157286942, −4.85860068794577940660273861650, −3.45012709302007568987307312375, −2.79291456173214772248973211945, −1.61696308614142262885175431345, −0.11307431033662277385962161195,
0.11307431033662277385962161195, 1.61696308614142262885175431345, 2.79291456173214772248973211945, 3.45012709302007568987307312375, 4.85860068794577940660273861650, 5.49802359397779796042157286942, 6.75292872494400020860570808565, 7.53998433678162739301622800888, 7.85171910890825771439941783451, 8.268326236063130669308822971407
